Method and apparatus for training a system model with gain constraints using a non-linear programming optimizer

ABSTRACT

Method and apparatus for training a system model with gain constraints. A method is disclosed for training a steady-state model, the model having an input and an output and a mapping layer for mapping the input to the output through a stored representation of a system. A training data set is provided having a set of input data u(t) and target output data y(t) representative of the operation of a system. The model is trained with a predetermined training algorithm which is constrained to maintain the sensitivity of the output with respect to the input substantially within user defined constraint bounds by iteratively minimizing an objective function as a function of a data objective and a constraint objective. The data objective has a data fitting learning rate and the constraint objective has constraint learning rate that are varied as a function of the values of the data objective and the constraint objective after selective iterative steps.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. patent application Ser. No.09/662,243 entitled “Method and Apparatus for Training a System Modelwith Gain Constraints”, filed on Sep. 14, 2000 now U.S. Pat. No.7,058,617, whose inventors are Eric J. Hartman, Stephen Piche and MarkGerules, which claims benefit of priority of Provisional PatentApplication Ser. No. 60/153,791 entitled “Method and Apparatus forTraining a System Model with Gain Constraints”, filed Sep. 14, 1999,whose inventors are Eric J. Hartman, Stephen Piche and Mark Gerules, andwhich is a Continuation-in-Part Application of U.S. patent applicationSer. No. 09/167,504 entitled “Method for On-line Optimization of aPlant”, filed Oct. 6, 1998, whose inventors are Stephen Piche, John P.Havener and Donald Semrad, which is now U.S. Pat. No. 6,278,899, whichis a Continuation-in-Part Application of U.S. patent application Ser.No. 08/943,489 entitled “Method for Steady-State Identification Basedupon Identified Dynamics”, filed Oct. 3, 1997, whose inventors areStephen Piche, James David Keeler, Eric Hartman, William D. Johnson,Mark Gerules and Kadir Liano, which is now U.S. Pat. No. 6,047,221,which is a Continuation-in-Part of U.S. patent Ser. No. 08/643,464entitled “Method and Apparatus for Dynamic and Steady State Modelingover a Desired Path Between Two End Points”, filed May 6, 1996, whoseinventors are Gregory D. Martin, Eugene Boe, Stephen Piche, James DavidKeeler, Douglas Timmer, Mark Gerules and John P. Havener, which is nowU.S. Pat. No. 5,933,345.

TECHNICAL FIELD OF THE INVENTION

The present invention pertains in general to neural network basedcontrol systems and, more particularly, to on-line optimization thereof.

BACKGROUND OF THE INVENTION

Process models that are utilized for prediction, control andoptimization can be divided into two general categories, steady-statemodels and dynamic models. In each case the model is a mathematicalconstruct that characterizes the process, and process measurements areutilized to parameterize or fit the model so that it replicates thebehavior of the process. The mathematical model can then be implementedin a simulator for prediction or inverted by an optimization algorithmfor control or optimization.

Steady-state or static models are utilized in modern process controlsystems that usually store a great deal of data, this data typicallycontaining steady-state information at many different operatingconditions. The steady-state information is utilized to train anon-linear model wherein the process input variables are represented bythe vector U that is processed through the model to output the dependentvariable Y. The non-linear model is a steady-state phenomenological orempirical model developed utilizing several ordered pairs (U_(i), Y_(i))of data from different measured steady states. If a model is representedas:Y=P(U,Y)  (1)

where P is some parameterization, then the steady-state modelingprocedure can be presented as:({right arrow over (U)},{right arrow over (Y)})→P  (2)where U and Y are vectors containing the U_(i), Y_(i) ordered pairelements. Given the model P, then the steady-state process gain can becalculated as:

$\begin{matrix}{K = \frac{\Delta\;{P\left( {U,Y} \right)}}{\Delta\; U}} & (3)\end{matrix}$The steady-state model therefore represents the process measurementsthat are taken when the system is in a “static” mode. These measurementsdo not account for the perturbations that exist when changing from onesteady-state condition to another steady-state condition. This isreferred to as the dynamic part of a model.

A dynamic model is typically a linear model and is obtained from processmeasurements which are not steady-state measurements; rather, these arethe data obtained when the process is moved from one steady-statecondition to another steady-state condition. This procedure is where aprocess input or manipulated variable u(t) is input to a process with aprocess output or controlled variable y(t) being output and measured.Again, ordered pairs of measured data (u(I), y(I)) can be utilized toparameterize a phenomenological or empirical model, this time the datacoming from non-steady-state operation. The dynamic model is representedas:y(t)=p(u(t)y(t))  (4)where p is some parameterization. Then the dynamic modeling procedurecan be represented as:({right arrow over (u)},{right arrow over (y)})→p  (5)Where u and y are vectors containing the (u(I),y(I)) ordered pairelements. Given the model p, then the steady-state gain of a dynamicmodel can be calculated as:

$\begin{matrix}{k = \frac{\Delta\;{p\left( {u,y} \right)}}{\Delta\; u}} & (6)\end{matrix}$Unfortunately, almost always the dynamic gain k does not equal thesteady-state gain K, since the steady-state gain is modeled on a muchlarger set of data, whereas the dynamic gain is defined around a set ofoperating conditions wherein an existing set of operating conditions aremildly perturbed. This results in a shortage of sufficient non-linearinformation in the dynamic data set in which non-linear information iscontained within the static model. Therefore, the gain of the system maynot be adequately modeled for an existing set of steady-state operatingconditions. Thus, when considering two independent models, one for thesteady-state model and one for the dynamic model, there is a mis-matchbetween the gains of the two models when used for prediction, controland optimization. The reason for this mis-match are that thesteady-state model is non-linear and the dynamic model is linear, suchthat the gain of the steady-state model changes depending on the processoperating point, with the gain of the linear model being fixed. Also,the data utilized to parameterize the dynamic model do not represent thecomplete operating range of the process, i.e., the dynamic data is onlyvalid in a narrow region. Further, the dynamic model represents theacceleration properties of the process (like inertia) whereas thesteady-state model represents the tradeoffs that determine the processfinal resting value (similar to the tradeoff between gravity and dragthat determines terminal velocity in free fall).

One technique for combining non-linear static models and linear dynamicmodels is referred to as the Hammerstein model. The Hammerstein model isbasically an input-output representation that is decomposed into twocoupled parts. This utilizes a set of intermediate variables that aredetermined by the static models which are then utilized to construct thedynamic model. These two models are not independent and are relativelycomplex to create.

Plants have been modeled utilizing the various non-linear networks. Onetype of network that has been utilized in the past is a neural network.These neural networks typically comprise a plurality of inputs which aremapped through a stored representation of the plant to yield on theoutput thereof predicted outputs. These predicted outputs can be anyoutput of the plant. The stored representation within the plant istypically determined through a training operation.

During the training of a neural network, the neural network is presentedwith a set of training data. This training data typically compriseshistorical data taken from a plant. This historical data is comprised ofactual input data and actual output data, which output data is referredto as the target data. During training, the actual input data ispresented to the network with the target data also presented to thenetwork, and then the network trained to reduce the error between thepredicted output from the network and the actual target data. One verywidely utilized technique for training a neural network is abackpropagation training algorithm. However, there are other types ofalgorithms that can be utilized to set the “weights” in the network.

When a large amount of steady-state data is available to a network, thestored representation can be accurately modeled. However, some plantshave a large amount of dynamic information associated therewith. Thisdynamic information reflects the fact that the inputs to the plantundergo a change which results in a corresponding change in the output.If a user desired to predict the final steady-state value of the plant,plant dynamics may not be important and this data could be ignored.However, there are situations wherein the dynamics of the plant areimportant during the prediction. It may be desirable to predict the paththat an output will take from a beginning point to an end point. Forexample, if the input were to change in a step function from one valueto another, a steady-state model that was accurately trained wouldpredict the final steady-state value with some accuracy. However, thepath between the starting point and the end point would not bepredicted, as this would be subject to the dynamics of the plant.Further, in some control applications, it may be desirable to actuallycontrol the plant such that the plant dynamics were “constrained,” thisrequiring some knowledge of the dynamic operation of the plant.

In some applications, the actual historical data that is available asthe training set has associated therewith a considerable amount ofdynamic information. If the training data set had a large amount ofsteady-state information, an accurate model could easily be obtained fora steady-state model. However, if the historical data had a large amountof dynamic information associated therewith, i.e., the plant were notallowed to come to rest for a given data point, then there would be anerror associated with the training operation that would be a result ofthis dynamic component in the training data. This is typically the casefor small data sets. This dynamic component must therefore be dealt withfor small training data sets when attempting to train a steady-statemodel.

When utilizing a model for the purpose of optimization, it is necessaryto train a model on one set of input values to predict another set ofinput values at future time. This will typically require a steady-statemodeling technique. In optimization, especially when used in conjunctionwith a control system, the optimization process will take a desired setof set points and optimizes those set points. However, these models aretypically selected for accurate gain a problem arises whenever theactual plant changes due to external influences, such as outsidetemperature, build up of slag, etc. Of course, one could regenerate themodel with new parameters. However, the typical method is to actuallymeasure the output of the plant, compare it with a predicted value togenerate a “biased” value which sets forth the error in the plant asopposed to the model. This error is then utilized to bias theoptimization network. However, to date this technique has required theuse of steady-state models which are generally off-line models. Thereason for this is that the actual values must “settle out” to reach asteady-state value before the actual bias can be determined. Duringoperation of a plant, the outputs are dynamic.

SUMMARY OF THE INVENTION

The present invention disclosed and claimed herein comprises a methodfor training a steady-state model, the model having an input and anoutput and a mapping layer for mapping the input to the output through astored representation of a system. A training data set is providedhaving a set of input data u(t) and target output data y(t)representative of the operation of a system. The model is trained with apredetermined training algorithm. The training algorithm is constrainedto maintain the sensitivity of the output with respect to the inputsubstantially within user defined constraint bounds by iterativelyminimizing an objective function as a function of a data objective and aconstraint objective. The data objective has a data fitting learningrate and the constraint objective has constraint learning rate. The datafitting learning rate and the constraint learning rate are varied as afunction of the values of the data objective and the constraintobjective after selective iterative steps.

In another embodiment, the best set of input variables is determinedfrom a set of historical training data for a system comprised of inputvariables and output variables in order to train a model of the systemon a smaller subset of the training data with less than all of the inputvariables. The best time delays are determined between input and outputvariables pairs in the training data prior to training of a modelthereon and a statistical relationship defined for each of the inputvariables at the best time delay for a given one of the outputvariables. Less than all of the input variables are selected by defininga maximum statistical relationship and selecting only those inputvariables for the given output variable having a statisticalrelationship that exceeds the maximum. The model is trained on theselected input variables and the associated output variables.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention and theadvantages thereof, reference is now made to the following descriptiontaken in conjunction with the accompanying Drawings in which:

FIG. 1 illustrates a prior art Hammerstein model;

FIG. 2 illustrates a block diagram of a modeling technique utilizingsteady-state gain to define the gain of the dynamic model;

FIGS. 3 a-3 d illustrate timing diagrams for the various outputs of thesystem of FIG. 2;

FIG. 4 illustrates a detailed block diagram of a dynamic model;

FIG. 5 illustrates a block diagram of the operation of the model of FIG.4;

FIG. 6 illustrates an example of the modeling technique of theembodiment of FIG. 2 utilized in a control environment;

FIG. 7 illustrates a diagrammatic view of a change between twosteady-state values;

FIG. 8 illustrates a diagrammatic view of the approximation algorithmfor changes in the steady-state value;

FIG. 9 illustrates a block diagram of the dynamic model;

FIG. 10 illustrates a detail of the control network utilizing an errorconstraining algorithm;

FIGS. 11 a and 11 b illustrate plots of the input and output duringoptimization;

FIG. 12 illustrates a plot depicting desired and predicted behavior;

FIG. 13 illustrates various plots for controlling a system to force thepredicted behavior to the desired behavior;

FIG. 14 illustrates a plot of a trajectory weighting algorithm;

FIG. 15 illustrates a plot for one type of constraining algorithm;

FIG. 16 illustrates a plot of the error algorithm as a function of time;

FIG. 17 illustrates a flowchart depicting the statistical method forgenerating the filter and defining the end point for the constrainingalgorithm of FIG. 15;

FIG. 18 illustrates a diagrammatic view of the optimization process;

FIG. 19 illustrates a flowchart for the optimization procedure;

FIG. 20 illustrates a diagrammatic view of the input space and the errorassociated therewith;

FIG. 21 illustrates a diagrammatic view of the confidence factor in theinput space;

FIG. 22 illustrates a block diagram of the method for utilizing acombination of a non-linear system and a first principals system;

FIG. 23 illustrates an alternate embodiment of the embodiment of FIG.22;

FIG. 24 illustrates a plot of a pair of data with a defined delayassociated therewith;

FIG. 25 illustrates a diagrammatic view of the binning method fordetermining statistical independence;

FIG. 26 illustrates a block diagram of a training method wherein delayis determined by statistical analysis;

FIG. 27 illustrates a flow chart of the method for determining delaysbased upon statistical methods;

FIGS. 27 a-27 c illustrate plots of the best input variable selectionprocess;

FIG. 27 d illustrates a scatterplot of dots that represent data of pHversus acid for a stirred-tank reactor;

FIG. 27 e illustrates a screen that allows the modeler to specify, foreach pair of input-output variables, gain constraints;

FIG. 27 f illustrates a screen wherein the modeler can selectinput-output cells;

FIG. 27 g illustrates the Gain Constraints Monitor viewable during andafter training a gain-constrained model;

FIG. 27 h illustrates a plot of gain constraints;

FIG. 28 illustrates a prior art Weiner model;

FIG. 29 illustrates a block diagram of a training method utilizing thesystem dynamics;

FIG. 30 illustrates plots of input data, actual output data, and thefiltered input data which has the plant dynamics impressed thereupon;

FIG. 31 illustrates a flow chart for the training operation;

FIG. 32 illustrates a diagrammatic view of the step test;

FIG. 33 illustrates a diagrammatic view of a single step for u(t) andû(t);

FIG. 34 illustrates a diagrammatic view of the pre-filter operationduring training;

FIG. 35 illustrates a diagrammatic view of a MIMO implementation of thetraining method of the present disclosure; and

FIG. 36 illustrates a non-fully connected network;

FIG. 37 illustrates a graphical user interface for selecting ranges ofvalues for the dynamic inputs in order to train the dynamic model;

FIG. 38 illustrates a flowchart depicting the selection of data andtraining of the model;

FIGS. 39 and 40 illustrate graphical user interfaces for depicting boththe actual historical response and the predictive response;

FIG. 41 illustrates a block diagram of a predictive control system witha GUI interface;

FIGS. 42-45 illustrate screen views for changing the number of variablesthat can be displayed from a given set;

FIG. 46 illustrates a diagrammatic view of a plant utilizing on-lineoptimization;

FIG. 47 illustrates a block diagram of the optimizer;

FIG. 48 illustrates a plot of manipulatable variables and controlledvariables or outputs;

FIGS. 49-51 illustrate plots of the dynamic operation of the system andthe bias;

FIG. 52 illustrates a block diagram of a prior art optimizer utilizingsteady-state;

FIG. 53 illustrates a diagrammatic view for determining the computeddisturbance variables;

FIG. 54 illustrates a block diagram for a steady state model utilizingthe computer disturbance variables;

FIG. 55 illustrates an overall block diagram of an optimization circuitutilizing computed disturbance variables;

FIG. 56 illustrates a diagrammatic view of furnace/boiler system whichhas associated therewith multiple levels of coal firing;

FIG. 57 illustrates a supply top sectional view of the tangentiallyfired furnace;

FIG. 58 illustrates a block diagram of one application of the on-lineoptimizer;

FIG. 59 illustrates a block diagram of training algorithm for training amodel using a multiple to single MV algorithm; and

FIGS. 60 and 61 illustrate more detailed block diagrams of theembodiment of FIG. 57.

DETAILED DESCRIPTION OF THE INVENTION

Referring now to FIG. 1, there is illustrated a diagrammatic view of aHammerstein model of the prior art. This is comprised of a non-linearstatic operator model 10 and a linear dynamic model 12, both disposed ina series configuration. The operation of this model is described in H.T. Su, and T. J. McAvoy, “Integration of Multilayer Perceptron Networksand Linear Dynamic Models: A Hammerstein Modeling Approach” to appear inI & EC Fundamentals, paper dated Jul. 7, 1992, which reference isincorporated herein by reference. Hammerstein models in general havebeen utilized in modeling non-linear systems for some time. Thestructure of the Hammerstein model illustrated in FIG. 1 utilizes thenon-linear static operator model 10 to transform the input U intointermediate variables H. The non-linear operator is usually representedby a finite polynomial expansion. However, this could utilize a neuralnetwork or any type of compatible modeling system. The linear dynamicoperator model 12 could utilize a discreet dynamic transfer functionrepresenting the dynamic relationship between the intermediate variableH and the output Y. For multiple input systems, the non-linear operatorcould utilize a multilayer neural network, whereas the linear operatorcould utilize a two layer neural network. A neural network for thestatic operator is generally well known and described in U.S. Pat. No.5,353,207, issued Oct. 4, 1994, and assigned to the present assignee,which is incorporated herein by reference. These type of networks aretypically referred to as a multilayer feed-forward network whichutilizes training in the form of back-propagation. This is typicallyperformed on a large set of training data. Once trained, the network hasweights associated therewith, which are stored in a separate database.

Once the steady-state model is obtained, one can then choose the outputvector from the hidden layer in the neural network as the intermediatevariable for the Hammerstein model. In order to determine the input forthe linear dynamic operator, u(t), it is necessary to scale the outputvector h(d) from the non-linear static operator model 10 for the mappingof the intermediate variable h(t) to the output variable of the dynamicmodel y(t), which is determined by the linear dynamic model.

During the development of a linear dynamic model to represent the lineardynamic operator, in the Hammerstein model, it is important that thesteady-state non-linearity remain the same. To achieve this goal, onemust train the dynamic model subject to a constraint so that thenon-linearity learned by the steady-state model remains unchanged afterthe training. This results in a dependency of the two models on eachother.

Referring now to FIG. 2, there is illustrated a block diagram of themodeling method in one embodiment, which is referred to as a systematicmodeling technique. The general concept of the systematic modelingtechnique in the present embodiment results from the observation that,while process gains (steady-state behavior) vary with U's and Y's,(i.e., the gains are non-linear), the process dynamics seemingly varywith time only, (i.e., they can be modeled as locally linear, buttime-varied). By utilizing non-linear models for the steady-statebehavior and linear models for the dynamic behavior, several practicaladvantages result. They are as follows:

-   -   1. Completely rigorous models can be utilized for the        steady-state part. This provides a credible basis for economic        optimization.    -   2. The linear models for the dynamic part can be updated        on-line, i.e., the dynamic parameters that are known to be        time-varying can be adapted slowly.    -   3. The gains of the dynamic models and the gains of the        steady-state models can be forced to be consistent (k=K).

With further reference to FIG. 2, there are provided a static orsteady-state model 20 and a dynamic model 22. The static model 20, asdescribed above, is a rigorous model that is trained on a large set ofsteady-state data. The static model 20 will receive a process input Uand provide a predicted output Y. These are essentially steady-statevalues. The steady-state values at a given time are latched in variouslatches, an input latch 24 and an output latch 26. The latch 24 containsthe steady-state value of the input U_(ss), and the latch 26 containsthe steady-state output value Y_(ss). The dynamic model 22 is utilizedto predict the behavior of the plant when a change is made from asteady-state value of Y_(ss) to a new value Y. The dynamic model 22receives on the input the dynamic input value u and outputs a predicteddynamic value y. The value u is comprised of the difference between thenew value U and the steady-state value in the latch 24, U_(ss). This isderived from a subtraction circuit 30 which receives on the positiveinput thereof the output of the latch 24 and on the negative inputthereof the new value of U. This therefore represents the delta changefrom the steady-state. Similarly, on the output the predicted overalldynamic value will be the sum of the output value of the dynamic model,y, and the steady-state output value stored in the latch 26, Y_(ss).These two values are summed with a summing block 34 to provide apredicted output Y. The difference between the value output by thesumming junction 34 and the predicted value output by the static model20 is that the predicted value output by the summing junction 20accounts for the dynamic operation of the system during a change. Forexample, to process the input values that are in the input vector U bythe static model 20, the rigorous model, can take significantly moretime than running a relatively simple dynamic model. The method utilizedin the present embodiment is to force the gain of the dynamic model 22k_(d) to equal the gain K_(ss) of the static model 20.

In the static model 20, there is provided a storage block 36 whichcontains the static coefficients associated with the static model 20 andalso the associated gain value K_(ss). Similarly, the dynamic model 22has a storage area 38 that is operable to contain the dynamiccoefficients and the gain value k_(d). One of the important aspects ofthe present embodiment is a link block 40 that is operable to modify thecoefficients in the storage area 38 to force the value of k_(d) to beequal to the value of K_(ss). Additionally, there is an approximationblock 41 that allows approximation of the dynamic gain k_(d) between themodification updates.

Systematic Model

The linear dynamic model 22 can generally be represented by thefollowing equations:

$\begin{matrix}{{\delta\;{y(t)}} = {{\sum\limits_{i = 1}^{n}{b_{i}\delta\;{u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}{a_{i}\delta\;{y\left( {t - i} \right)}}}}} & (7)\end{matrix}$where:δy(t)=y(t)−Y _(ss)  (8)δu(t)=u(t)−u _(ss)  (9)and t is time, a_(i) and b_(i) are real numbers, d is a time delay, u(t)is an input and y(t) an output. The gain is represented by:

$\begin{matrix}{\frac{y(B)}{u(B)} = {k = \frac{\left( {\sum\limits_{i = 1}^{n}{b_{i}B^{i - 1}}} \right)B^{d}}{1 + {\sum\limits_{i = 1}^{n}{a_{i}B^{i - 1}}}}}} & (10)\end{matrix}$where B is the backward shift operator B(x(t))=x(t−1), t=time, the a_(i)and b_(i) are real numbers, I is the number of discreet time intervalsin the dead-time of the process, and n is the order of the model. Thisis a general representation of a linear dynamic model, as contained inGeorge E. P. Box and G. M. Jenkins, “TIME SERIES ANALYSIS forecastingand control”, Holden-Day, San Francisco, 1976, Section 10.2, Page 345.This reference is incorporated herein by reference.

The gain of this model can be calculated by setting the value of B equalto a value of “1”. The gain will then be defined by the followingequation:

$\begin{matrix}{\left\lbrack \frac{y(B)}{u(B)} \right\rbrack_{B = 1} = {k_{d} = \frac{\sum\limits_{i = 1}^{n}b_{i}}{1 + {\sum\limits_{i = 1}^{n}a_{i}}}}} & (11)\end{matrix}$

The a_(i) contain the dynamic signature of the process, its unforced,natural response characteristic. They are independent of the processgain. The b_(i) contain part of the dynamic signature of the process;however, they alone contain the result of the forced response. The b_(i)determine the gain k of the dynamic model. See: J. L. Shearer, A. T.Murphy, and H. H. Richardson, “Introduction to System Dynamics”,Addison-Wesley, Reading, Mass., 1967, Chapter 12. This reference isincorporated herein by reference.

Since the gain K_(ss) of the steady-state model is known, the gain k_(d)of the dynamic model can be forced to match the gain of the steady-statemodel by scaling the b_(i) parameters. The values of the static anddynamic gains are set equal with the value of b_(i) scaled by the ratioof the two gains:

$\begin{matrix}{\left( b_{i} \right)_{scaled} = {\left( b_{i} \right)_{old}\left( \frac{K_{ss}}{k_{d}} \right)}} & (12)\end{matrix}$

$\begin{matrix}{\left( b_{1} \right)_{scaled} = \frac{\left( b_{i} \right)_{old}{K_{ss}\left( {1 + {\sum\limits_{I = 1}^{n}a_{1}}} \right)}}{\sum\limits_{i = 1}^{n}b_{1}}} & (13)\end{matrix}$This makes the dynamic model consistent with its steady-statecounterpart. Therefore, each time the steady-state value changes, thiscorresponds to a gain K_(ss) of the steady-state model. This value canthen be utilized to update the gain k_(d) of the dynamic model and,therefore, compensate for the errors associated with the dynamic modelwherein the value of k_(d) is determined based on perturbations in theplant on a given set of operating conditions. Since all operatingconditions are not modeled, the step of varying the gain will accountfor changes in the steady-state starting points.

Referring now to FIGS. 3 a-3 d, there are illustrated plots of thesystem operating in response to a step function wherein the input valueU changes from a value of 100 to a value of 110. In FIG. 3 a, the valueof 100 is referred to as the previous steady-state value U_(ss). In FIG.3 b, the value of u varies from a value of 0 to a value of 10, thisrepresenting the delta between the steady-state value of U_(ss) to thelevel of 110, represented by reference numeral 42 in FIG. 3 a.Therefore, in FIG. 3 b the value of u will go from 0 at a level 44, to avalue of 10 at a level 46. In FIG. 3 c, the output Y is represented ashaving a steady-state value Y_(ss) of 4 at a level 48. When the inputvalue U rises to the level 42 with a value of 110, the output value willrise. This is a predicted value. The predicted value which is the properoutput value is represented by a level 50, which level 50 is at a valueof 5. Since the steady-state value is at a value of 4, this means thatthe dynamic system must predict a difference of a value of 1. This isrepresented by FIG. 3 d wherein the dynamic output value y varies from alevel 54 having a value of 0 to a level 56 having a value of 1.0.However, without the gain scaling, the dynamic model could, by way ofexample, predict a value for y of 1.5, represented by dashed level 58,if the steady-state values were outside of the range in which thedynamic model was trained. This would correspond to a value of 5.5 at alevel 60 in the plot of FIG. 3 c. It can be seen that the dynamic modelmerely predicts the behavior of the plant from a starting point to astopping point, not taking into consideration the steady-state values.It assumes that the steady-state values are those that it was trainedupon. If the gain k_(d) were not scaled, then the dynamic model wouldassume that the steady-state values at the starting point were the samethat it was trained upon. However, the gain scaling link between thesteady-state model and the dynamic model allow the gain to be scaled andthe parameter b_(i) to be scaled such that the dynamic operation isscaled and a more accurate prediction is made which accounts for thedynamic properties of the system.

Referring now to FIG. 4, there is illustrated a block diagram of amethod for determining the parameters a_(i), b_(i). This is usuallyachieved through the use of an identification algorithm, which isconventional. This utilizes the (u(t),y(t)) pairs to obtain the a_(i)and b_(i) parameters. In the preferred embodiment, a recursiveidentification method is utilized where the a_(i) and b_(i) parametersare updated with each new (u_(i)(t),y_(i)(t)) pair. See: T. Eykhoff,“System Identification”, John Wiley & Sons, New York, 1974, Pages 38 and39, et. seq., and H. Kurz and W. Godecke, “Digital Parameter-AdaptiveControl Processes with Unknown Dead Time”, Automatica, Vol. 17, No. 1,1981, pp. 245-252, which references are incorporated herein byreference.

In the technique of FIG. 4, the dynamic model 22 has the output thereofinput to a parameter-adaptive control algorithm block 60 which adjuststhe parameters in the coefficient storage block 38, which also receivesthe scaled values of k, b_(i). This is a system that is updated on aperiodic basis, as defined by timing block 62. The control algorithm 60utilizes both the input u and the output y for the purpose ofdetermining and updating the parameters in the storage area 38.

Referring now to FIG. 5, there is illustrated a block diagram of thepreferred method. The program is initiated in a block 68 and thenproceeds to a function block 70 to update the parameters a_(i), b_(i)utilizing the (u(I),y(I)) pairs. Once these are updated, the programflows to a function block 72 wherein the steady-state gain factor K isreceived, and then to a function block 74 to set the dynamic gain to thesteady-state gain, i.e., provide the scaling function describedhereinabove. This is performed after the update. This procedure can beused for on-line identification, non-linear dynamic model prediction andadaptive control.

Referring now to FIG. 6, there is illustrated a block diagram of oneapplication of the present embodiment utilizing a control environment. Aplant 78 is provided which receives input values u(t) and outputs anoutput vector y(t). The plant 78 also has measurable state variabless(t). A predictive model 80 is provided which receives the input valuesu(t) and the state variables s(t) in addition to the output value y(t).The steady-state model 80 is operable to output a predicted value ofboth y(t) and also of a future input value u(t+1). This constitutes asteady-state portion of the system. The predicted steady-state inputvalue is U_(ss) with the predicted steady-state output value beingY_(ss). In a conventional control scenario, the steady-state model 80would receive as an external input a desired value of the outputy^(d)(t) which is the desired value that the overall control systemseeks to achieve. This is achieved by controlling a distributed controlsystem (DCS) 86 to produce a desired input to the plant. This isreferred to as u(t+1), a future value. Without considering the dynamicresponse, the predictive model 80, a steady-state model, will providethe steady-state values. However, when a change is desired, this changewill effectively be viewed as a “step response”.

To facilitate the dynamic control aspect, a dynamic controller 82 isprovided which is operable to receive the input u(t), the output valuey(t) and also the steady-state values U_(ss) and Y_(ss) and generate theoutput u(t+1). The dynamic controller effectively generates the dynamicresponse between the changes, i.e., when the steady-state value changesfrom an initial steady-state value U_(ss) ^(i), Y^(i) _(ss) to a finalsteady-state value U^(f) _(ss), Y^(f) _(ss).

During the operation of the system, the dynamic controller 82 isoperable in accordance with the embodiment of FIG. 2 to update thedynamic parameters of the dynamic controller 82 in a block 88 with again link block 90, which utilizes the value K_(ss) from a steady-stateparameter block in order to scale the parameters utilized by the dynamiccontroller 82, again in accordance with the above described method. Inthis manner, the control function can be realized. In addition, thedynamic controller 82 has the operation thereof optimized such that thepath traveled between the initial and final steady-state values isachieved with the use of the optimizer 83 in view of optimizerconstraints in a block 85. In general, the predicted model (steady-statemodel) 80 provides a control network function that is operable topredict the future input values. Without the dynamic controller 82, thisis a conventional control network which is generally described in U.S.Pat. No. 5,353,207, issued Oct. 4, 1994, to the present assignee, whichpatent is incorporated herein by reference.

Approximate Systematic Modeling

For the modeling techniques described thus far, consistency between thesteady-state and dynamic models is maintained by rescaling the b_(i)parameters at each time step utilizing equation 13. If the systematicmodel is to be utilized in a Model Predictive Control (MPC) algorithm,maintaining consistency may be computationally expensive. These types ofalgorithms are described in C. E. Garcia, D. M. Prett and M. Morari.Model predictive control: theory and practice—a survey, Automatica,25:335-348, 1989; D. E. Seborg, T. F. Edgar, and D. A. Mellichamp.Process Dynamics and Control. John Wiley and Sons, New York, N.Y., 1989.These references are incorporated herein by reference. For example, ifthe dynamic gain k_(d) is computed from a neural network steady-statemodel, it would be necessary to execute the neural network module eachtime the model was iterated in the MPC algorithm. Due to the potentiallylarge number of model iterations for certain MPC problems, it could becomputationally expensive to maintain a consistent model. In this case,it would be better to use an approximate model which does not rely onenforcing consistencies at each iteration of the model.

Referring now to FIG. 7, there is illustrated a diagram for a changebetween steady-state values. As illustrated, the steady-state model willmake a change from a steady-state value at a line 100 to a steady-statevalue at a line 102. A transition between the two steady-state valuescan result in unknown settings. The only way to insure that the settingsfor the dynamic model between the two steady-state values, an initialsteady-state value K_(ss) ^(i) and a final steady-state gain K_(ss)^(f), would be to utilize a step operation, wherein the dynamic gaink_(d) was adjusted at multiple positions during the change. However,this may be computationally expensive. As will be described hereinbelow,an approximation algorithm is utilized for approximating the dynamicbehavior between the two steady-state values utilizing a quadraticrelationship. This is defined as a behavior line 104, which is disposedbetween an envelope 106, which behavior line 104 will be describedhereinbelow.

Referring now to FIG. 8, there is illustrated a diagrammatic view of thesystem undergoing numerous changes in steady-state value as representedby a stepped line 108. The stepped line 108 is seen to vary from a firststeady-state value at a level 110 to a value at a level 112 and thendown to a value at a level 114, up to a value at a level 116 and thendown to a final value at a level 118. Each of these transitions canresult in unknown states. With the approximation algorithm that will bedescribed hereinbelow, it can be seen that, when a transition is madefrom level 110 to level 112, an approximation curve for the dynamicbehavior 120 is provided. When making a transition from level 114 tolevel 116, an approximation gain curve 124 is provided to approximatethe steady-state gains between the two levels 114 and 116. For makingthe transition from level 116 to level 118, an approximation gain curve126 for the steady-state gain is provided. It can therefore be seen thatthe approximation curves 120-126 account for transitions betweensteady-state values that are determined by the network, it being notedthat these are approximations which primarily maintain the steady-stategain within some type of error envelope, the envelope 106 in FIG. 7.

The approximation is provided by the block 41 noted in FIG. 2 and can bedesigned upon a number of criteria, depending upon the problem that itwill be utilized to solve. The system in the preferred embodiment, whichis only one example, is designed to satisfy the following criteria:

-   -   1. Computational Complexity: The approximate systematic model        will be used in a Model Predictive Control algorithm, therefore,        it is required to have low computational complexity.    -   2. Localized Accuracy: The steady-state model is accurate in        localized regions. These regions represent the steady-state        operating regimes of the process. The steady-state model is        significantly less accurate outside these localized regions.    -   3. Final Steady-State: Given a steady-state set point change, an        optimization algorithm which uses the steady-state model will be        used to compute the steady-state inputs required to achieve the        set point. Because of item 2, it is assumed that the initial and        final steady-states associated with a set-point change are        located in regions accurately modeled by the steady-state model.

Given the noted criteria, an approximate systematic model can beconstructed by enforcing consistency of the steady-state and dynamicmodel at the initial and final steady-state associated with a set pointchange and utilizing a linear approximation at points in between the twosteady-states. This approximation guarantees that the model is accuratein regions where the steady-state model is well known and utilizes alinear approximation in regions where the steady-state model is known tobe less accurate. In addition, the resulting model has low computationalcomplexity. For purposes of this proof, Equation 13 is modified asfollows:

$\begin{matrix}{b_{i,{scaled}} = \frac{b_{i}\;{K_{ss}\left( {u\left( {t - d - 1} \right)} \right)}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}{\sum\limits_{i = 1}^{n}b_{i}}} & (14)\end{matrix}$

This new equation 14 utilizes K_(ss)(u(t−d−1)) instead of K_(ss)(u(t))as the consistent gain, resulting in a systematic model which is delayinvariant.

The approximate systematic model is based upon utilizing the gainsassociated with the initial and final steady-state values of a set-pointchange. The initial steady-state gain is denoted K^(i) _(ss) while theinitial steady-state input is given by U^(i) _(ss). The finalsteady-state gain is K^(f) _(ss) and the final input is U^(f) _(ss).Given these values, a linear approximation to the gain is given by:

$\begin{matrix}{{K_{ss}\left( {u(t)} \right)} = {K_{ss}^{i} + {\frac{K_{ss}^{f} - K_{ss}^{i}}{U_{ss}^{f} - U_{ss}^{i}}{\left( {{u(t)} - U_{ss}^{i}} \right).}}}} & (15)\end{matrix}$Substituting this approximation into Equation 13 and replacingu(t−d−1)−u^(i) by δu(t−d−1) yields:

$\begin{matrix}\begin{matrix}{{\overset{\sim}{b}}_{j,{scaled}} = {\frac{b_{j}{K_{ss}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}} + \frac{1}{2}}} \\{\frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}\delta\;{{u\left( {t - d - i} \right)}.}}\end{matrix} & (16)\end{matrix}$To simplify the expression, define the variable b_(j)-Bar as:

$\begin{matrix}{{\overset{\_}{b}}_{j} = \frac{b_{j}{K_{ss}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}}} & (17)\end{matrix}$and g_(j) as:

$\begin{matrix}{g_{j} = \frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}} & (18)\end{matrix}$Equation 16 may be written as:{tilde over (b)} _(j,scaled) = b _(j) +g _(j) δu(t−d−i).  (19)Finally, substituting the scaled b's back into the original differenceEquation 7, the following expression for the approximate systematicmodel is obtained:

$\begin{matrix}{{\delta\;{y(t)}} = {{\sum\limits_{i = 1}^{n}{{\overset{\_}{b}}_{i}\delta\;{u\left( {t - d - i} \right)}}} + {\sum\limits_{i = 1}^{n}{g_{i}\delta\;{u\left( {t - d - i^{2}} \right)}\delta\;{u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}{a_{i}\delta\;{y\left( {t - i} \right)}}}}} & (20)\end{matrix}$The linear approximation for gain results in a quadratic differenceequation for the output. Given Equation 20, the approximate systematicmodel is shown to be of low computational complexity. It may be used ina MPC algorithm to efficiently compute the required control moves for atransition from one steady-state to another after a set-point change.Note that this applies to the dynamic gain variations betweensteady-state transitions and not to the actual path values.Control System Error Constraints

Referring now to FIG. 9, there is illustrated a block diagram of theprediction engine for the dynamic controller 82 of FIG. 6. Theprediction engine is operable to essentially predict a value of y(t) asthe predicted future value y(t+1). Since the prediction engine mustdetermine what the value of the output y(t) is at each future valuebetween two steady-state values, it is necessary to perform these in a“step” manner. Therefore, there will be k steps from a value of zero toa value of N, which value at k=N is the value at the “horizon”, thedesired value. This, as will be described hereinbelow, is an iterativeprocess, it being noted that the terminology for “(t+1)” refers to anincremental step, with an incremental step for the dynamic controllerbeing smaller than an incremented step for the steady-state model. Forthe steady-state model, “y(t+N)” for the dynamic model will be, “y(t+1)”for the steady state The value y(t+1) is defined as follows:y(t+1)=a ₁ y(t)+a ₂ y(t−1)+b ₁ u(t−d−1)+b ₂ u(t−d−2)  (21)With further reference to FIG. 9, the input values u(t) for each (u,y)pair are input to a delay line 140. The output of the delay lineprovides the input value u(t) delayed by a delay value “d”. There areprovided only two operations for multiplication with the coefficients b₁and b₂, such that only two values u(t) and u(t−1) are required. Theseare both delayed and then multiplied by the coefficients b₁ and b₂ andthen input to a summing block 141. Similarly, the output value y^(p)(t)is input to a delay line 142, there being two values required formultiplication with the coefficients a₁ and a₂. The output of thismultiplication is then input to the summing block 141. The input to thedelay line 142 is either the actual input value y^(a)(t) or the iteratedoutput value of the summation block 141, which is the previous valuecomputed by the dynamic controller 82. Therefore, the summing block 141will output the predicted value y(t+1) which will then be input to amultiplexor 144. The multiplexor 144 is operable to select the actualoutput y^(a)(t) on the first operation and, thereafter, select theoutput of the summing block 141. Therefore, for a step value of k=0 thevalue y^(a)(t) will be selected by the multiplexor 144 and will belatched in a latch 145. The latch 145 will provide the predicted valuey^(p)(t+k) on an output 146. This is the predicted value of y(t) for agiven k that is input back to the input of delay line 142 formultiplication with the coefficients a₁ and a₂. This is iterated foreach value of k from k=0 to k=N.

The a₁ and a₂ values are fixed, as described above, with the b₁ and b₂values scaled. This scaling operation is performed by the coefficientmodification block 38. However, this only defines the beginningsteady-state value and the final steady-state value, with the dynamiccontroller and the optimization routines described in the presentapplication defining how the dynamic controller operates between thesteady-state values and also what the gain of the dynamic controller is.The gain specifically is what determines the modification operationperformed by the coefficient modification block 38.

In FIG. 9, the coefficients in the coefficient modification block 38 aremodified as described hereinabove with the information that is derivedfrom the steady-state model. The steady-state model is operated in acontrol application, and is comprised in part of a forward steady-statemodel 141 which is operable to receive the steady-state input valueU_(ss)(t) and predict the steady-state output value Y_(ss)(t). Thispredicted value is utilized in an inverse steady-state model 143 toreceive the desired value y^(d)(t) and the predicted output of thesteady-state model 141 and predict a future steady-state input value ormanipulated value U_(ss)(t+N) and also a future steady-state input valueY_(ss)(t+N) in addition to providing the steady-state gain K_(ss). Asdescribed hereinabove, these are utilized to generate scaled b-values.These b-values are utilized to define the gain k_(d) of the dynamicmodel. In can therefore be seen that this essentially takes a lineardynamic model with a fixed gain and allows it to have a gain thereofmodified by a non-linear model as the operating point is moved throughthe output space.

Referring now to FIG. 10, there is illustrated a block diagram of thedynamic controller and optimizer. The dynamic controller includes adynamic model 149 which basically defines the predicted value y^(p)(k)as a function of the inputs y(t), s(t) and u(t). This was essentiallythe same model that was described hereinabove with reference to FIG. 9.The model 149 predicts the output values y^(p)(k) between the twosteady-state values, as will be described hereinbelow. The model 149 ispredefined and utilizes an identification algorithm to identify the a₁,a₂, b₁ and b₂ coefficients during training. Once these are identified ina training and identification procedure, these are “fixed”. However, asdescribed hereinabove, the gain of the dynamic model is modified byscaling the coefficients b₁ and b₂. This gain scaling is not describedwith respect to the optimization operation of FIG. 10, although it canbe incorporated in the optimization operation.

The output of model 149 is input to the negative input of a summingblock 150. Summing block 150 sums the predicted output y^(p)(k) with thedesired output y^(d)(t). In effect, the desired value of y^(d)(t) iseffectively the desired steady-state value Y^(f) _(ss), although it canbe any desired value. The output of the summing block 150 comprises anerror value which is essentially the difference between the desiredvalue y^(d)(t) and the predicted value y^(p)(k). The error value ismodified by an error modification block 151, as will be describedhereinbelow, in accordance with error modification parameters in a block152. The modified error value is then input to an inverse model 153,which basically performs an optimization routine to predict a change inthe input value u(t). In effect, the optimizer 153 is utilized inconjunction with the model 149 to minimize the error output by summingblock 150. Any optimization function can be utilized, such as a MonteCarlo procedure. However, in the present embodiment, a gradientcalculation is utilized. In the gradient method, the gradient ∂(y)/∂(u)is calculated and then a gradient solution performed as follows:

$\begin{matrix}{{\Delta\; u_{new}} = {{\Delta\; u_{old}} + {\left( \frac{\partial(y)}{\partial(u)} \right) \times E}}} & (22)\end{matrix}$

The optimization function is performed by the inverse model 153 inaccordance with optimization constraints in a block 154. An iterationprocedure is performed with an iterate block 155 which is operable toperform an iteration with the combination of the inverse model 153 andthe predictive model 149 and output on an output line 156 the futurevalue u(t+k+1). For k=0, this will be the initial steady-state value andfor k=N, this will be the value at the horizon, or at the nextsteady-state value. During the iteration procedure, the previous valueof u(t+k) has the change value Δu added thereto. This value is utilizedfor that value of k until the error is within the appropriate levels.Once it is at the appropriate level, the next u(t+k) is input to themodel 149 and the value thereof optimized with the iterate block 155.Once the iteration procedure is done, it is latched. As will bedescribed hereinbelow, this is a combination of modifying the error suchthat the actual error output by the block 150 is not utilized by theoptimizer 153 but, rather, a modified error is utilized. Alternatively,different optimization constraints can be utilized, which are generatedby the block 154, these being described hereinbelow.

Referring now to FIGS. 11 a and 11 b, there are illustrated plots of theoutput y(t+k) and the input u_(k)(t+k+1), for each k from the initialsteady-state value to the horizon steady-state value at k=N. Withspecific reference to FIG. 11 a, it can be seen that the optimizationprocedure is performed utilizing multiple passes. In the first pass, theactual value u^(a)(t+k) for each k is utilized to determine the valuesof y(t+k) for each u,y pair. This is then accumulated and the valuesprocessed through the inverse model 153 and the iterate block 155 tominimize the error. This generates a new set of inputs u_(k)(t+k+1)illustrated in FIG. 11 b. Therefore, the optimization after pass 1generates the values of u(t+k+1) for the second pass. In the secondpass, the values are again optimized in accordance with the variousconstraints to again generate another set of values for u(t+k+1). Thiscontinues until the overall objective function is reached. Thisobjective function is a combination of the operations as a function ofthe error and the operations as a function of the constraints, whereinthe optimization constraints may control the overall operation of theinverse model 153 or the error modification parameters in block 152 maycontrol the overall operation. Each of the optimization constraints willbe described in more detail hereinbelow.

Referring now to FIG. 12, there is illustrated a plot of y^(d)(t) andy^(p)(t). The predicted value is represented by a waveform 170 and thedesired output is represented by a waveform 172, both plotted over thehorizon between an initial steady-state value Y^(i) _(ss) and a finalsteady-state value Y^(f) _(ss). It can be seen that the desired waveformprior to k=0 is substantially equal to the predicted output. At k=0, thedesired output waveform 172 raises its level, thus creating an error. Itcan be seen that at k=0, the error is large and the system then mustadjust the manipulated variables to minimize the error and force thepredicted value to the desired value. The objective function for thecalculation of error is of the form:

$\begin{matrix}{\min\limits_{\Delta\; u_{il}}{\sum\limits_{j}{\sum\limits_{k}\left( {A_{j}*\left( {{{\overset{\rightarrow}{y}}^{p}(t)} - {{\overset{\rightarrow}{y}}^{d}(t)}} \right)^{2}} \right.}}} & (23)\end{matrix}$where:

Du_(il) is the change in input variable (IV) I at time interval l

A_(j) is the weight factor for control variable (CV) j

y^(p)(t) is the predicted value of CV j at time interval k

y^(d)(t) is the desired value of CV j.

Trajectory Weighting

The present system utilizes what is referred to as “trajectoryweighting” which encompasses the concept that one does not put aconstant degree of importance on the future predicted process behaviormatching the desired behavior at every future time set, i.e., at lowk-values. One approach could be that one is more tolerant of error inthe near term (low k-values) than farther into the future (highk-values). The basis for this logic is that the final desired behavioris more important than the path taken to arrive at the desired behavior,otherwise the path traversed would be a step function. This isillustrated in FIG. 13 wherein three possible predicted behaviors areillustrated, one represented by a curve 174 which is acceptable, one isrepresented by a different curve 176, which is also acceptable and onerepresented by a curve 178, which is unacceptable since it goes abovethe desired level on curve 172. Curves 174-178 define the desiredbehavior over the horizon for k=1 to N.

In Equation 23, the predicted curves 174-178 would be achieved byforcing the weighting factors A_(j) to be time varying. This isillustrated in FIG. 14. In FIG. 14, the weighting factor A as a functionof time is shown to have an increasing value as time and the value of kincreases. This results in the errors at the beginning of the horizon(low k-values) being weighted much less than the errors at the end ofthe horizon (high k-values). The result is more significant than merelyredistributing the weights out to the end of the control horizon at k=N.This method also adds robustness, or the ability to handle a mismatchbetween the process and the prediction model. Since the largest error isusually experienced at the beginning of the horizon, the largest changesin the independent variables will also occur at this point. If there isa mismatch between the process and the prediction (model error), theseinitial moves will be large and somewhat incorrect, which can cause poorperformance and eventually instability. By utilizing the trajectoryweighting method, the errors at the beginning of the horizon areweighted less, resulting in smaller changes in the independent variablesand, thus, more robustness.

Error Constraints

Referring now to FIG. 15, there are illustrated constraints that can beplaced upon the error. There is illustrated a predicted curve 180 and adesired curve 182, desired curve 182 essentially being a flat line. Itis desirable for the error between curve 180 and 182 to be minimized.Whenever a transient occurs at t=0, changes of some sort will berequired. It can be seen that prior to t=0, curve 182 and 180 aresubstantially the same, there being very little error between the two.However, after some type of transition, the error will increase. If arigid solution were utilized, the system would immediately respond tothis large error and attempt to reduce it in as short a time aspossible. However, a constraint frustum boundary 184 is provided whichallows the error to be large at t=0 and reduces it to a minimum level ata point 186. At point 186, this is the minimum error, which can be setto zero or to a non-zero value, corresponding to the noise level of theoutput variable to be controlled. This therefore encompasses the sameconcepts as the trajectory weighting method in that final futurebehavior is considered more important that near term behavior. The evershrinking minimum and/or maximum bounds converge from a slack positionat t=0 to the actual final desired behavior at a point 186 in theconstraint frustum method.

The difference between constraint frustum and trajectory weighting isthat constraint frustums are an absolute limit (hard constraint) whereany behavior satisfying the limit is just as acceptable as any otherbehavior that also satisfies the limit. Trajectory weighting is a methodwhere differing behaviors have graduated importance in time. It can beseen that the constraints provided by the technique of FIG. 15 requiresthat the value y^(p)(t) is prevented from exceeding the constraintvalue. Therefore, if the difference between y^(d)(t) and y^(p)(t) isgreater than that defined by the constraint boundary, then theoptimization routine will force the input values to a value that willresult in the error being less than the constraint value. In effect,this is a “clamp” on the difference between y^(p)(t) and y^(d)(t). Inthe trajectory weighting method, there is no “clamp” on the differencetherebetween; rather, there is merely an attenuation factor placed onthe error before input to the optimization network.

Trajectory weighting can be compared with other methods, there being twomethods that will be described herein, the dynamic matrix control (DMC)algorithm and the identification and command (IdCom) algorithm. The DMCalgorithm utilizes an optimization to solve the control problem byminimizing the objective function:

$\begin{matrix}{{\min\limits_{\Delta\; U_{il}}{\sum\limits_{j}{\sum\limits_{k}{\text{(}A_{j}*\left( {{{\overset{\rightarrow}{y}}^{P}(t)} - {{\overset{\rightarrow}{y}}^{D}(t)}} \right)}}}} + {\sum\limits_{i}{B_{i}*{\sum\limits_{1}\left( {\Delta\; U_{il}} \right)^{2}}}}} & (24)\end{matrix}$where B_(i) is the move suppression factor for input variable I. This isdescribed in Cutler, C. R. and B. L. Ramaker, Dynamic Matrix Control—AComputer Control Algorithm, AIChE National Meeting, Houston, Tex.(April, 1979), which is incorporated herein by reference.

It is noted that the weights A_(j) and desired values y^(d)(t) areconstant for each of the control variables. As can be seen from Equation24, the optimization is a trade off between minimizing errors betweenthe control variables and their desired values and minimizing thechanges in the independent variables. Without the move suppression term,the independent variable changes resulting from the set point changeswould be quite large due to the sudden and immediate error between thepredicted and desired values. Move suppression limits the independentvariable changes, but for all circumstances, not just the initialerrors.

The IdCom algorithm utilizes a different approach. Instead of a constantdesired value, a path is defined for the control variables to take fromthe current value to the desired value. This is illustrated in FIG. 16.This path is a more gradual transition from one operation point to thenext. Nevertheless, it is still a rigidly defined path that must be met.The objective function for this algorithm takes the form:

$\begin{matrix}{\min\limits_{\Delta\; U_{il}}{\sum\limits_{j}{\sum\limits_{k}\left( {A_{j}*\left( {Y^{P_{jk}} - y_{refjk}} \right)} \right)^{2}}}} & (25)\end{matrix}$This technique is described in Richalet, J., A. Rault, J. L. Testud, andJ. Papon, Model Predictive Heuristic Control: Applications to IndustrialProcesses, Automatica, 14, 413-428 (1978), which is incorporated hereinby reference. It should be noted that the requirement of Equation 25 ateach time interval is sometimes difficult. In fact, for controlvariables that behave similarly, this can result in quite erraticindependent variable changes due to the control algorithm attempting toendlessly meet the desired path exactly.

Control algorithms such as the DMC algorithm that utilize a form ofmatrix inversion in the control calculation, cannot handle controlvariable hard constraints directly. They must treat them separately,usually in the form of a steady-state linear program. Because this isdone as a steady-state problem, the constraints are time invariant bydefinition. Moreover, since the constraints are not part of a controlcalculation, there is no protection against the controller violating thehard constraints in the transient while satisfying them at steady-state.

With further reference to FIG. 15, the boundaries at the end of theenvelope can be defined as described hereinbelow. One techniquedescribed in the prior art, W. Edwards Deming, “Out of the Crisis,”Massachusetts Institute of Technology, Center for Advanced EngineeringStudy, Cambridge Mass., Fifth Printing, September 1988, pages 327-329,describes various Monte Carlo experiments that set forth the premisethat any control actions taken to correct for common process variationactually may have a negative impact, which action may work to increasevariability rather than the desired effect of reducing variation of thecontrolled processes. Given that any process has an inherent accuracy,there should be no basis to make a change based on a difference thatlies within the accuracy limits of the system utilized to control it. Atpresent, commercial controllers fail to recognize the fact that changesare undesirable, and continually adjust the process, treating alldeviation from target, no matter how small, as a special cause deservingof control actions, i.e., they respond to even minimal changes. Overadjustment of the manipulated variables therefore will result, andincrease undesirable process variation. By placing limits on the errorwith the present filtering algorithms described herein, only controlleractions that are proven to be necessary are allowed, and thus, theprocess can settle into a reduced variation free from unmeritedcontroller disturbances. The following discussion will deal with onetechnique for doing this, this being based on statistical parameters.

Filters can be created that prevent model-based controllers from takingany action in the case where the difference between the controlledvariable measurement and the desired target value are not significant.The significance level is defined by the accuracy of the model uponwhich the controller is statistically based. This accuracy is determinedas a function of the standard deviation of the error and a predeterminedconfidence level. The confidence level is based upon the accuracy of thetraining. Since most training sets for a neural network-based model willhave “holes” therein, this will result in inaccuracies within the mappedspace. Since a neural network is an empirical model, it is only asaccurate as the training data set. Even though the model may not havebeen trained upon a given set of inputs, it will extrapolate the outputand predict a value given a set of inputs, even though these inputs aremapped across a space that is questionable. In these areas, theconfidence level in the predicted output is relatively low. This isdescribed in detail in U. S. patent application Ser. No. 08/025,184,filed Mar. 2, 1993, which is incorporated herein by reference.

Referring now to FIG. 17, there is illustrated a flowchart depicting thestatistical method for generating the filter and defining the end point186 in FIG. 15. The flowchart is initiated at a start block 200 and thenproceeds to a function block 202, wherein the control values u(t+1) arecalculated. However, prior to acquiring these control values, thefiltering operation must be a processed. The program will flow to afunction block 204 to determine the accuracy of the controller. This isdone off-line by analyzing the model predicted values compared to theactual values, and calculating the standard deviation of the error inareas where the target is undisturbed. The model accuracy of e_(m)(t) isdefined as follows:e _(m)(t)=a(t)−p(t)  (26)

where:

-   -   e_(m)=model error,    -   a=actual value    -   p=model predicted value        The model accuracy is defined by the following equation:        Acc=H*σ _(m)  (27)

where:

-   -   Acc=accuracy in terms of minimal detector error    -   H=significance level=1 67% confidence        -   =2 95% confidence        -   =3 99.5% confidence    -   σ_(m)=standard deviation of e_(m)(t).        The program then flows to a function block 206 to compare the        controller error e_(c)(t) with the model accuracy. This is done        by taking the difference between the predicted value (measured        value) and the desired value. This is the controller error        calculation as follows:        e _(c)(t)=d(t)−m(t)  (28)

where:

-   -   e_(c)=controller error    -   d=desired value    -   m=measured value        The program will then flow to a decision block 208 to determine        if the error is within the accuracy limits. The determination as        to whether the error is within the accuracy limits is done        utilizing Shewhart limits. With this type of limit and this type        of filter, a determination is made as to whether the controller        error e_(c)(t) meets the following conditions: e_(c)(t)≧−1*Acc        and e_(c)(t)≦+1*Acc, then either the control action is        suppressed or not suppressed. If it is within the accuracy        limits, then the control action is suppressed and the program        flows along a “Y” path. If not, the program will flow along the        “N” path to function block 210 to accept the u(t+1) values. If        the error lies within the controller accuracy, then the program        flows along the “Y” path from decision block 208 to a function        block 212 to calculate the running accumulation of errors. This        is formed utilizing a CUSUM approach. The controller CUSUM        calculations are done as follows:        S _(low)=min(0,S _(low)(t−1)+d(t)−m(t))−Σ(m)+k)  (29)        S _(hi)=max(0,S _(hi)(t−1)+[d(t)−m(t))−Σ(m)]−k)  (30)

where:

-   -   S_(hi)=Running Positive Qsum    -   S_(low)=Running Negative Qsum    -   k=Tuning factor−minimal detectable change threshold with the        following defined:    -   Hq=significance level. Values of (j,k) can be found so that the        CUSUM control chart will have significance levels equivalent to        Shewhart control charts.        The program will then flow to a decision block 214 to determine        if the CUSUM limits check out, i.e., it will determine if the        Qsum values are within the limits. If the Qsum, the accumulated        sum error, is within the established limits, the program will        then flow along the “Y” path. And, if it is not within the        limits, it will flow along the “N” path to accept the controller        values u(t+1). The limits are determined if both the value of        S_(hi)≧+1*Hq and S_(low)≦−1*Hq. Both of these actions will        result in this program flowing along the “Y” path. If it flows        along the “N” path, the sum is set equal to zero and then the        program flows to the function block 210. If the Qsum values are        within the limits, it flows along the “Y” path to a function        block 218 wherein a determination is made as to whether the user        wishes to perturb the process. If so, the program will flow        along the “Y” path to the function block 210 to accept the        control values u(t+1). If not, the program will flow along the        “N” path from decision block 218 to a function block 222 to        suppress the controller values u(t+1). The decision block 218,        when it flows along the “Y” path, is a process that allows the        user to re-identify the model for on-line adaptation, i.e.,        retrain the model. This is for the purpose of data collection        and once the data has been collected, the system is then        reactivated.

Referring now to FIG. 18, there is illustrated a block diagram of theoverall optimization procedure. In the first step of the procedure, theinitial steady-state values {Y_(ss) ^(i), U_(ss) ^(i)} and the finalsteady-state values {Y_(ss) ^(f), U_(ss) ^(f)} are determined, asdefined in blocks 226 and 228, respectively. In some calculations, boththe initial and the final steady-state values are required. The initialsteady-state values are utilized to define the coefficients a^(i), b^(i)in a block 228. As described above, this utilizes the coefficientscaling of the b-coefficients. Similarly, the steady-state values inblock 228 are utilized to define the coefficients a^(f), b^(f), it beingnoted that only the b-coefficients are also defined in a block 229. Oncethe beginning and end points are defined, it is then necessary todetermine the path therebetween. This is provided by block 230 for pathoptimization. There are two methods for determining how the dynamiccontroller traverses this path. The first, as described above, is todefine the approximate dynamic gain over the path from the initial gainto the final gain. As noted above, this can incur some instabilities.The second method is to define the input values over the horizon fromthe initial value to the final value such that the desired value Y_(ss)^(f) is achieved. Thereafter, the gain can be set for the dynamic modelby scaling the b-coefficients. As noted above, this second method doesnot necessarily force the predicted value of the output y^(p)(t) along adefined path; rather, it defines the characteristics of the model as afunction of the error between the predicted and actual values over thehorizon from the initial value to the final or desired value. Thiseffectively defines the input values for each point on the trajectoryor, alternatively, the dynamic gain along the trajectory.

The steady-state model is operable to predict both the outputsteady-state value Y_(ss) ^(i) at a value of k=0, the initialsteady-state value, and the output steady-state value Y_(ss) ^(i) at atime t+N where k=N, the final steady-state value. At the initialsteady-state value, there is defined a region which comprises a surfacein the output space in the proximity of the initial steady-state value,which initial steady-state value also lies in the output space. Thisdefines the range over which the dynamic controller can operate and therange over which it is valid. At the final steady-state value, if thegain were not changed, the dynamic model would not be valid. However, byutilizing the steady-state model to calculate the steady-state gain atthe final steady-state value and then force the gain of the dynamicmodel to equal that of the steady-state model, the dynamic model thenbecomes valid over a region proximate the final steady-state value. Thisis at a value of k=N. The problem that arises is how to define the pathbetween the initial and final steady-state values. One possibility, asmentioned hereinabove, is to utilize the steady-state model to calculatethe steady-state gain at multiple points along the path between theinitial steady-state value and the final steady-state value and thendefine the dynamic gain at those points. This could be utilized in anoptimization routine, which could require a large number ofcalculations. If the computational ability were there, this wouldprovide a continuous calculation for the dynamic gain along the pathtraversed between the initial steady-state value and the finalsteady-state value utilizing the steady-state gain. However, it ispossible that the steady-state model is not valid in regions between theinitial and final steady-state values, i.e., there is a low confidencelevel due to the fact that the training in those regions may not beadequate to define the model therein. Therefore, the dynamic gain isapproximated in these regions, the primary goal being to have someadjustment of the dynamic model along the path between the initial andthe final steady-state values during the optimization procedure. Thisallows the dynamic operation of the model to be defined.

Referring now to FIG. 19, there is illustrated a flow chart depictingthe optimization algorithm. The program is initiated at a start block232 and then proceeds to a function block 234 to define the actual inputvalues u^(a)(t) at the beginning of the horizon, this typically beingthe steady-state value U_(ss). The program then flows to a functionblock 235 to generate the predicted values y^(p)(k) over the horizon forall k for the fixed input values. The program then flows to a functionblock 236 to generate the error E(k) over the horizon for all k for thepreviously generated y^(p)(k). These errors and the predicted values arethen accumulated, as noted by function block 238. The program then flowsto a function block 240 to optimize the value of u(t) for each value ofk in one embodiment. This will result in k-values for u(t). Of course,it is sufficient to utilize less calculations than the totalk-calculations over the horizon to provide for a more efficientalgorithm. The results of this optimization will provide the predictedchange Δu(t+k) for each value of k in a function block 242. The programthen flows to a function block 243 wherein the value of u(t+k) for eachu will be incremented by the value Δu(t+k). The program will then flowto a decision block 244 to determine if the objective function notedabove is less than or equal to a desired value. If not, the program willflow back along an “N” path to the input of function block 235 to againmake another pass. This operation was described above with respect toFIGS. 11 a and 11 b. When the objective function is in an acceptablelevel, the program will flow from decision block 244 along the “Y” pathto a function block 245 to set the value of u(t+k) for all u. Thisdefines the path. The program then flows to an End block 246.

Steady State Gain Determination

Referring now to FIG. 20, there is illustrated a plot of the input spaceand the error associated therewith. The input space is comprised of twovariables x₁ and x₂. The y-axis represents the function f(x₁, x₂). Inthe plane of x₁ and x₂, there is illustrated a region 250, whichrepresents the training data set. Areas outside of the region 250constitute regions of no data, i.e., a low confidence level region. Thefunction Y will have an error associated therewith. This is representedby a plane 252. However, the error in the plane 250 is only valid in aregion 254, which corresponds to the region 250. Areas outside of region254 on plane 252 have an unknown error associated therewith. As aresult, whenever the network is operated outside of the region 250 withthe error region 254, the confidence level in the network is low. Ofcourse, the confidence level will not abruptly change once outside ofthe known data regions but, rather, decreases as the distance from theknown data in the training set increases. This is represented in FIG. 21wherein the confidence is defined as α(x). It can be seen from FIG. 21that the confidence level α(x) is high in regions overlying the region250.

Once the system is operating outside of the training data regions, i.e.,in a low confidence region, the accuracy of the neural net is relativelylow. In accordance with one aspect of the preferred embodiment, a firstprinciples model g(x) is utilized to govern steady-state operation. Theswitching between the neural network model f(x) and the first principlemodels g(x) is not an abrupt switching but, rather, it is a mixture ofthe two.

The steady-state gain relationship is defined in Equation 7 and is setforth in a more simple manner as follows:

$\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {f\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (31)\end{matrix}$A new output function Y(u) is defined to take into account theconfidence factor α(u) as follows:Y({right arrow over (u)})=α({right arrow over (u)})·f({right arrow over(u)})+(1−α({right arrow over (u)}))g({right arrow over (u)})  (32)

where:

-   -   α(u)=confidence in model f (u)    -   α(u) in the range of 0→1    -   α(u)ε{0,1}        This will give rise to the relationship:

$\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {Y\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (33)\end{matrix}$In calculating the steady-state gain in accordance with this Equationutilizing the output relationship Y(u), the following will result:

$\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = {{\frac{\partial\left( {\alpha\left( \;\overset{\rightarrow}{u} \right)} \right)}{\partial\left( \;\overset{\rightarrow}{u} \right)} \times {F\left( \overset{\rightarrow}{u} \right)}} + {{\alpha\left( \overset{\rightarrow}{u} \right)}\frac{\partial\left( {F\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} + {\frac{\partial\left( {1 - {\alpha\left( \;\overset{\rightarrow}{u} \right)}} \right)}{\partial\left( \;\overset{\rightarrow}{u} \right)} \times {g\left( \overset{\rightarrow}{u} \right)}} + {\left( {1 - {\alpha\left( \overset{\rightarrow}{u} \right)}} \right)\frac{\partial\left( {g\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}}}} & (34)\end{matrix}$

Referring now to FIG. 22, there is illustrated a block diagram of theembodiment for realizing the switching between the neural network modeland the first principles model. A neural network block 300 is providedfor the function f(u), a first principle block 302 is provided for thefunction g(u) and a confidence level block 304 for the function α(u).The input u(t) is input to each of the blocks 300-304. The output ofblock 304 is processed through a subtraction block 306 to generate thefunction 1−α(u), which is input to a multiplication block 308 formultiplication with the output of the first principles block 302. Thisprovides the function (1−α(u))*g(u). Additionally, the output of theconfidence block 304 is input to a multiplication block 310 formultiplication with the output of the neural network block 300. Thisprovides the function f(u)*α(u). The output of block 308 and the outputof block 310 are input to a summation block 312 to provide the outputY(u).

Referring now to FIG. 23, there is illustrated an alternate embodimentwhich utilizes discreet switching. The output of the first principlesblock 302 and the neural network block 300 are provided and are operableto receive the input x(t). The output of the network block 300 and firstprinciples block 302 are input to a switch 320, the switch 320 operableto select either the output of the first principals block 302 or theoutput of the neural network block 300. The output of the switch 320provides the output Y(u).

The switch 320 is controlled by a domain analyzer 322. The domainanalyzer 322 is operable to receive the input x(t) and determine whetherthe domain is one that is within a valid region of the network 300. Ifnot, the switch 320 is controlled to utilize the first principlesoperation in the first principles block 302. The domain analyzer 322utilizes the training database 326 to determine the regions in which thetraining data is valid for the network 300. Alternatively, the domainanalyzer 320 could utilize the confidence factor α(u) and compare thiswith a threshold, below which the first principles model 302 would beutilized.

Identification of Dynamic Models

Gain information, as noted hereinabove, can also be utilized in thedevelopment of dynamic models. Instead of utilizing the user-specifiedgains, the gains may be obtained from a trained steady-state model.Although described hereinabove with reference to Equation 7, a singleinput, single output dynamic model will be defined by a similar equationas follows:ŷ(t)=−a ₁ ŷ(t−1)−a ₂ ŷ(t−2)+b ₁ u(t−d−1)+b ₂ u(t−d−2)  (35)where the dynamic steady-state gain is defined as follows:

$\begin{matrix}{k_{d} = \frac{b_{1} + b_{2}}{1 + a_{1} + a_{2}}} & (36)\end{matrix}$This gain relationship is essentially the same as defined hereinabove inEquation 11. Given a time series of input and output data, u(t) andy(t), respectively, and the steady-state or static gain associated withthe average value of the input, K′_(ss), the parameters of the dynamicsystem may be defined by minimizing the following cost function:

$\begin{matrix}{J = {{\lambda\left( {k_{d} - K_{ss}^{\prime}} \right)}^{2} + {\sum\limits_{t = t_{i}}^{\;_{t_{f}}}\left( {{y^{p}(t)} - {y(t)}} \right)^{2}}}} & (37)\end{matrix}$where λ is a user-specified value. It is noted that the second half ofEquation 37 constitutes the summation over the time series with thevalue y(t) constituting the actual output and the function y^(p)(t)constituting the predicted output values. The mean square error of thisterm is summed from an initial time t_(i) to a final time t_(f),constituting the time series. The gain value k_(d) basically constitutesthe steady-state gain of the dynamic model. This optimization is subjectto the following constraints on dynamic stability:0≦a₂<1  (38)−a ₂−1<a ₁<0  (39)−a ₂−1<a ₁<0  (39)which are conventional constraints. The variable λ is used to enforcethe average steady-state gain, K′_(ss), in the identification of thedynamic model. The value K′_(ss) is found by calculating the averagevalue of the steady-state gain associated with the neural network overthe time horizon t_(i) to t_(f). Given the input time series u(t_(i)) tou(t_(f)), the K′_(ss) is defined as follows:

$\begin{matrix}{K_{ss}^{\prime} = {\frac{1}{t_{f} - t_{i}}{\sum\limits_{t = t_{i}}^{t_{f}}{K_{ss}(t)}}}} & (40)\end{matrix}$For a large value of λ, the gain of the steady-state and dynamic modelsare forced to be equal. For a small value of λ, the gain of the dynamicmodel is found independently from that of the steady-state model. Forλ=0, the optimization problem is reduced to a technique commonlyutilized in identification of output equation-based models, as definedin L. Ljung, “System Identification: Theory for the User,”Prentice-Hall, Englewood Cliffs, N.J. 1987.

In defining the dynamic model in accordance with Equation No. 37, it isrecognized that only three parameters need to be optimized, the a₁parameter, the a₂ parameter and the ratio of b₁ and b₂. This is to becompared with the embodiment described hereinabove with reference toFIG. 2, wherein the dynamic gain was forced to be equal to thesteady-state gain of the static model 20. By utilizing the weightingfactor λ and minimizing the cost function in accordance with Equation 37without requiring the dynamic gain k_(d) to equal the steady-state gainK′_(ss) of the neural network, some latitude is provided in identifyingthe dynamic model.

In the embodiment described above with respect to FIG. 2, the model wasidentified by varying the b-values with the dynamic gain forced to beequal to the steady-state gain. In the embodiment illustrated above withrespect to Equation 37, the dynamic gain does not necessarily have toequal the steady-state gain K′_(ss), depending upon the value of λdefining the weighting factor.

The above noted technique of Equation 37 provides for determining thea's and b's of the dynamic model as a method of identification in aparticular localized region of the input space. Once the a's and b's ofthe dynamic model are known, this determines the dynamics of the systemwith the only variation over the input space from the localized regionin which the dynamic step test data was taken being the dynamic gaink_(d). If this gain is set to a value of one, then the only componentremaining are the dynamics. Therefore, the dynamic model, once defined,then has its gain scaled to a value of one, which merely requiresadjusting the b-values. This will be described hereinbelow. Afteridentification of the model, it is utilized as noted hereinabove withrespect to the embodiment of FIG. 2 and the dynamic gain can then bedefined for each region utilizing the static gain.

Steady-State Model Identification

As noted hereinabove, to optimize and control any process, a model ofthat process is needed. The present system relies on a combination ofsteady-state and dynamic models. The quality of the model determines theoverall quality of the final control of the plant. Various techniquesfor training a steady-state model will be described hereinbelow.

Prior to discussing the specific model identification method utilized inthe present embodiment, it is necessary to define a quasi-steady-statemodel. For comparison, a steady-state model will be defined as follows:

Steady-State Models:

A steady-state model is represented by the static mapping from theinput, u(t) to the output y(t) as defined by the following equation:y (t)=F(ū(t))  (41)where F(u(t)) models the steady-state mapping of the process andu(t)εR^(m) and y(t)εR^(n) represent the inputs and outputs of a givenprocess. It should be noted that the input, u(t), and the output, y(t),are not a function of time and, therefore, the steady-state mapping isindependent of time. The gain of the process must be defined withrespect to a point in the input space. Given the point ū(t), the gain ofprocess is defined as:

$\begin{matrix}{{G\left( {\overset{\rightharpoonup}{u}(t)} \right)} = {\frac{\mathbb{d}\overset{\rightharpoonup}{y}}{\mathbb{d}\overset{\rightharpoonup}{u}}❘_{u{(t)}}}} & (42)\end{matrix}$where G is a R^(m×n) matrix. This gain is equivalent to the sensitivityof the function F(u(t)).Quasi-Steady-State Models:

A steady-state model by definition contains no time information. In somecases, to identify steady-state models, it is necessary to introducetime information into the static model:y (t)=G(ū(t,d))  (43)where:ū(t,d)=[u ₁(t−d ₁)u ₂(t−d ₂) . . . u _(m)(t−d _(m))]  (44)The variable d_(i) represents the delay associated with the i^(th)input. In the quasi-steady-state model, the static mapping of G(u(t)) isessentially equal to the steady-state mapping F(u(t)). The response ofsuch a model is illustrated in FIG. 24. In FIG. 24, there is illustrateda single input u₁(t) and a single output y₁(t), this being a singleoutput, single input system. There is a delay, or dead time d, notedbetween the input and the output which represents a quasi-steady-statedynamic. It is noted, however, that each point on the input u₁(t)corresponds to a given point on the output y₁(t) by some delay d.However, when u₁(t) makes a change from an initial value to a finalvalue, the output makes basically an instantaneous change and followsit. With respect to the quasi-steady-state model, the only dynamics thatare present in this model is the delay component.Identification of Delays in Quasi-Steady-State Models

Given data generated by a quasi-steady-state model of the formy(t)=G({right arrow over (u)}(t−d)),  (45)where d is the dead-time or delay noted in FIG. 24, the generatingfunction G( ) is approximated via a neural network training algorithm(nonlinear regression) when d is known. That is, a function G( ) isfitted to a set of data points generated from G( ), where each datapoint is a u(t), y(t) pair. The present system concerns time-seriesdata, and thus the dataset is indexed by t. The data set is denoted byD.

In process modeling, exact values for d are ordinarily not critical tothe quality of the model; approximate values typically suffice. Priorart systems specified a method for approximating d by training a modelwith multiple delays per input, and picking the delay which has thelargest sensitivity (average absolute partial derivative of outputw.r.t. input). In these prior art systems, the sensitivity was typicallydetermined by manipulating a given input and determining the effectthereof on the output. By varying the delay, i.e., taking a differentpoint of data in time with respect to a given y(t) value, a measure ofsensitivity of the output on the input can be determined as a functionof the delay. By taking the delay which exhibits the largestsensitivity, the delay of the system can be determined.

The disadvantage to the sensitivity technique is that it requires anumber of passes through the network during training in order todetermine the delay. This is an iterative technique. In accordance withthe present system, the method for approximating the delay is doneutilizing a statistical method for examining the data, as will bedescribed in more detail hereinbelow. This method is performed withoutrequiring actual neural network training during the determinationoperation. The method of the present embodiment examines each inputvariable against a given output variable, independently of the otherinput variables. Given d_(i) for an input variable u_(i), the methodmeasures the strength of the relationship between u_(i)(t−d_(i)) andy(t). The method is fast, such that many d_(i) values may be tested foreach u_(i).

The user supplies d_(i,min) and d_(i,max) values for each u_(i). Thestrength of the relationship between u_(i)(t−d_(i)) and y(t) is computedfor each d_(i) between d_(i,min) and d_(i,max) (inclusive). The d_(i)yielding the strongest relationship between u_(i)(t−d_(i)) and y(t) ischosen as the approximation of the dead-time for that input variable onthe given output variable. The strength of the relationship betweenu_(i)(t−d_(i)) and y(t) is defined as the degree of statisticaldependence between u_(i)(t−d_(i)) and y(t). The degree of statisticaldependence between u_(i)(t−d_(i)) and y(t) is the degree to whichu_(i)(t−d_(i)) and y(t) are not statistically independent.

Statistical dependence is a general concept. As long as there is anyrelationship whatsoever between two variables, of whatever form, linearor nonlinear, the definition of statistical independence for those twovariables will fail. Statistical independence between two variablesx₁(t) and x₂(t) is defined as:p(x ₁(t))p(x ₂(t))=p(x ₁(t),x₂(t))∀t  (46)where p(x₁(t)) is the marginal probability density function of x₁(t) andp(x₁(t),x₂(t)) is the joint probability density function(x₁(t)=u_(i)(t−d_(j)) and x₂=y(t)); that is, the product of the marginalprobabilities is equal to the joint probability. If they are equal, thisconstitutes statistical independence, and the level of inequalityprovides a measure of statistical dependence.

Any measure f(x₁(t),x₂(t)) which has the following property (“Property1”) is a suitable measure of statistical dependence:

-   -   Property 1: f(x₁(t),x₂(t)) is 0 if and only if Equation 46 holds        at each data point, and f>0 otherwise. In addition, the        magnitude of f measures the degree of violation of Equation 46        summed over all data points.

Mutual information (MI) is one such measure, and is defined as:

$\begin{matrix}{{MI} = {\sum\limits_{t}{{p\left( {{x_{1}(t)},{x_{2}(t)}} \right)}{\log\left( \frac{p\left( {{x_{1}(t)},{x_{2}(t)}} \right)}{{p\left( {x_{1}(t)} \right)},{p\left( {x_{2}(t)} \right)}} \right)}}}} & (47)\end{matrix}$Property 1 holds for MI. Theoretically, there is no fixed maximum valueof MI, as it depends upon the distributions of the variables inquestion. As explained hereinbelow, the maximum, as a practical matter,also depends upon the method employed to estimate probabilities.Regardless, MI values for different pairs of variables may be ranked byrelative magnitude to determine the relative strength of therelationship between each pair of variables. Any other measure f havingProperty 1 would be equally applicable, such as the sum of the squaresof the product of the two sides of Equation 58:SSD=Σ _(l)[(p(x ₁(t),x ₂(t))−p(x ₁(t))p(x ₂(t)))]²  (48)However, MI (Equation 47) is the preferred method in the disclosedembodiment.Statistical Dependence vs. Correlation

For purposes of the present embodiment, the method described above,i.e., measuring statistical dependence, is superior to using linearcorrelation. The definition of linear correlation is well-known and isnot stated herein. Correlation ranges in value from −1 to 1, and itsmagnitude indicates the degree of linear relationship between variablesx₁(t) and x₂(t). Nonlinear relationships are not detected bycorrelation. For example, y(t) is totally determined by x(t) in therelationy(t)=x ²(t).  (49)Yet, if x(t) varies symmetrically about zero, then:corr(y(t),x(t))=0.  (50)That is, correlation detects no linear relationship because therelationship is entirely nonlinear. Conversely, statistical dependenceregisters a relationship of any kind, linear or nonlinear, betweenvariables. In this example, MI(y(t),x(t)) would calculate to be a largenumber.Estimation Probabilities

An issue in computing MI is how to estimate the probabilitydistributions given a dataset D. Possible methods include kernelestimation methods, and binning methods. The preferred method is abinning method, as binning methods are significantly cheaper to computethan kernel estimation methods.

Of the binning techniques, a very popular method is that disclosed in A.M. Fraser and Harry L. Swinney. “Independent Coordinates for StrangeAttractors in Mutual Information,” Physical Review A, 33(2):1134-1140,1986. This method makes use of a recursive quadrant-division process.

The present method uses a binning method whose performance is highlysuperior to that of the Fraser method. The binning method used hereinsimply divides each of the two dimensions (u_(i)(t−d_(j)) and y(t)) intoa fixed number of divisions, where N is a parameter which may besupplied by the user, or which defaults to sqrt(#datapoints/20). Thewidth of each division is variable, such that an (approximately) equalnumber of points fall into each division of the dimension. Thus, theprocess of dividing each dimension is independent of the otherdimension.

In order to implement the binning procedure, it is first necessary todefine a grid of data points for each input value at a given delay. Eachinput value will be represented by a time series and will therefore be aseries of values. For example, if the input value were u₁(t), therewould be a time series of these u₁(t) values, u₁(t₁), u₁(t₂) . . .u₁(t_(f)). There would be a time series u₁(t) for each output valuey(t). For the purposes of the illustration herein, there will beconsidered only a single output from y(t), although it should beunderstood that a multiple input, multiple output system could beutilized.

Referring now to FIG. 25, there is illustrated a diagrammatic view of abinning method. In this method, a single point generated for each valueof u_(i)(t) for the single value y(t). All of the data in the timeseries u_(i)(t) is plotted in a single grid. This time series is thendelayed by the delay value d_(j) to provide a delay valueu_(i)(t−d_(j)). For each value of j from d_(j,min) to d_(j,max), therewill be a grid generated. There will then be a mutual information valuegenerated for each grid to show the strength of the relationship betweenthat particular delay value d_(j) and the output y(t). By continuallychanging the delay d_(j) for the time series u_(i)(t), a different MIvalue can be generated.

In the illustration of FIG. 25, there are illustrated a plurality ofrows and a plurality columns with the data points disposed therein withthe x-axis labeled u_(i)(t−d_(j)) and the y-axis labeled y(t). For agiven column 352 and a given row 354, there is defined a single bin 350.As described above, the grid lines are variable such that the number ofpoints in any one division is variable, as described hereinabove. Oncethe grid is populated, then it is necessary to determine the MI value.This MI value for the binning grid or a given value of d_(j) is definedas follows:

$\begin{matrix}{{MI} = {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{M}{{p\left( {{x_{i}(i)},{x_{2}(j)}} \right)}\log\frac{p\left( {{x_{1}(i)},{x_{2}(j)}} \right)}{{p\left( {x_{1}(i)} \right)}{p\left( {x_{2}(j)} \right)}}}}}} & (51)\end{matrix}$where p(x₁(i),x₂(j)) is equal to the number of points in a particularbin over the total number of points in the grid, p(x₁(i)) is equal tothe number of points in a column over the total number of points andp(x₂(j)) is equal to the number of data points in a row over the totalnumber of data points in the grid and n is equal to the number of rowsand M is equal to the number of columns. Therefore, it can be seen thatif the data was equally distributed around the grid, the value of MIwould be equal to zero. As the strength of the relationship increases asa function of the delay value, then it would be noted that the pointstend to come together in a strong relationship, and the value of MIincreases. The delay d_(j) having the strongest relationship willtherefore be selected as the proper delay for that given u_(i)(t).

Referring now to FIG. 26, there is illustrated a block diagram depictingthe use of the statistical analysis approach. The statistical analysisis defined in a block 353 which receive both the values of y(t) andu(t). This statistical analysis is utilized to select for each u_(i)(t)the appropriate delay d_(j). This, of course, is for each y(t). Theoutput of this is stored in a delay register 355. During training of anon-linear neural network 357, a delay block 359 is provided forselecting from the data set of u(t) for given u_(i)(t) an appropriatedelay and introducing that delay into the value before inputting it to atraining block 358 for training the neural network 357. The trainingblock 358 also utilizes the data set for y(t) as target data. Again, theparticular delay for the purpose of training is defined by statisticalanalysis block 353, in accordance with the algorithms describedhereinabove.

Referring now to FIG. 27, there is illustrated a flow chart depictingthe binning operation. The procedure is initiated at a block 356 andproceeds to a block 360 to select a given one of the outputs y(t) for amulti-output system and then to a block 362 to select one of the inputvalues u_(i)(t−d_(j)). It then flows to a function block 363 to set thevalue of d_(j) to the minimum value and then to a block 364 to performthe binning operation wherein all the points for that particular delayd_(j) are placed onto the grid. The MI value is then calculated for thisgrid, as indicated by a block 365. The program then proceeds to adecision block 366 to determine if the value of d_(j) is equal to themaximum value d_(j,max). If not, this value is incremented by a block367 and then proceeds back to the input of block 364 to increment thenext delay value for u_(i)(t−d_(j)). This continues until the delay hasvaried from d_(i,min) through d_(j,max). The program then flows to thedecision block 368 to determine if there are additional input variables.If so, the program flows to a block 369 to select the next variable andthen back to the input of block 363. If not, the program flows to ablock 370 to select the next value of y(t). This will then flow back tothe input of function block 360 until all input variables and outputvariables have been processed. The program will then flow to an ENDblock 371.

Automatic Selection of Best Input Variable

It often occurs that many more input variables are available to themodeler than are necessary or sufficient to create a neural networkmodel that is as good or better than a model created using all of theavailable input variables. In this case, it is desirable to have amethod to assist the modeler to identify the best relatively smallsubset of the set of available input variables that would enable thecreation of a model whose quality is as good or better than (1) a modelthat uses the entire set of available input variables, and (2) a modelthat uses any other relatively small subset of the available inputvariables. That is, we want an algorithm to assist the modeler indetermining the best, hopefully relatively small, set of inputvariables, from the total set of input variables available (recognizingthat in some cases the total set of available input variables may be thebest set of input variables).

The criterion for choosing the “best” set of input variables thereforeinvolves a tradeoff between having fewer variables, and hence a smaller,more manageable model, and increasing the number of input variables butachieving diminishing returns in terms of model quality in the process.

The method utilizes the mutual information, as described hereinabove, todetermine the best time delays between input and output variables priorto training a neural network model. As described, for each pair ofinput-output variables, a mutual information value is obtained for eachtime delay increment between a minimum and maximum time delay specifiedby the modeler. For a given input-output pair, the maximum mutualinformation value across all of the time delays examined is chosen asthe best time delay.

Thus, each pair of input-output variables has an associated maximummutual information value. To assist the user in identifying the best setof input variables for a given output variable, the maximum mutualinformation values for all available input variables (corresponding tothat output variable) are plotted in sequence of decreasing maximummutual information values as shown in FIG. 27 a. In the plot, themaximum mutual information values for the ten total available inputvariables are shown on the vertical axis, and the input variable numberranked by decreasing maximum mutual information value is shown on thehorizontal axis. The user may identify the name of the variable byselecting the “Info” Tool, and clicking the mouse on any of the pointsin the plot, each of which represents one available input variable.

As shown in the plot of FIG. 27 b, the modeler can then make their owndetermination regarding the above-mentioned trade-off regarding thenumber of input variables versus the amount of additional informationdelivered to the model. By selecting the “Include Left”Tool, the usercan click anywhere on the plot to indicate how many of the left-mostvariables are to be included as inputs to the model. The left-mostvariables are those having the highest mutual information in associationwith the output variable indicated at the top right corner of the plot—“” ppm” in this case. The plot is displayable for each output in themodel. In the plot shown, the modeler has chosen to select seven of theten possible input variables for the model.

FIG. 27 c illustrates the appearance of the screen after the user clicksthe “Apply” button: the rejected three right-most variable are shownwith larger dots than the seven retained variables.

Identification of Steady-State Models Using Gain Constraints:

In most processes, bounds upon the steady-state gain are known eitherfrom the first principles or from practical experience. Once it isassumed that the gain information is known, a method for utilizing thisknowledge of empirically-based models will be described herein. If oneconsiders a parameterized quasi-steady-state model of the form:{right arrow over (y)}(t)= N ({right arrow over (w)},{right arrow over(u)}(t−d))  (52)where w is a vector of free parameters (typically referred to as theweights of a neural network) and N(w,u(t−d)) represents a continuousfunction of both w and u(t−d). A feedforward neural network as describedhereinabove represents an example of the nonlinear function. A commontechnique for identifying the free parameters w is to establish sometype of cost function and then minimize this cost function using avariety of different optimization techniques, including such techniquesas steepest descent or conjugate gradients. As an example, duringtraining of feedforward neural networks utilizing a backpropogationalgorithm, it is common to minimize the mean squared error over atraining set,

$\begin{matrix}{{J\left( \overset{\rightharpoonup}{w} \right)} = {\sum\limits_{t = 1}^{P}\left( {{\overset{\rightharpoonup}{y}(t)} - {{\overset{\rightharpoonup}{y}}_{d}(t)}} \right)^{2}}} & (53)\end{matrix}$where P is the number of training patterns, y^(d)(t) is the trainingdata or target data, y(t) is the predicted output and J(w) is the error.

Constraints upon the gains of steady-state models may be taken intoaccount in determining w by modifying the optimization problem. As notedabove, w is determined by establishing a cost function and thenutilizing an optimization technique to minimize the cost function. Gainconstraints may be introduced into the problem by specifying them aspart of the optimization problem. Thus, the optimization problem may bereformulated as:min(J({right arrow over (w)}))  (54)subject toG _(l)({right arrow over (u)}(1))<G({right arrow over (u)}(1))<G_(h)({right arrow over (u)}(1))  (55)G _(l)({right arrow over (u)}(2))<G({right arrow over (u)}(2))<G_(h)({right arrow over (u)}(2))  (56). . .  (57)G _(k)({right arrow over (u)}(P))<G(u(P))<G _(h)({right arrow over(u)}(P))  (58)where G_(l)(u(t)) is the matrix of the user-specified lower gainconstraints and G_(h)(u(t)) are the upper gain constraints. Each of thegain constraints represents the enforcement of a lower and upper gain ona single one of the input-output pairs of the training set, i.e., thegain is bounded for each input-output pair and can have a differentvalue. These are what are referred to as “hard constraints.” Thisoptimization problem may be solved utilizing a non-linear programmingtechnique.

Another approach to adding the constraints to the optimization problemis to modify the cost function, i.e., utilize some type of softconstraints. For example, the squared error cost function of Equation 53may be modified to account for the gain constraints in the gain asfollows:

$\begin{matrix}{{J(w)} = {{\sum\limits_{t = 1}^{P}\left( {{\overset{->}{y}(t)} - {{\overset{->}{y}}_{d}(t)}} \right)^{2}} + {\lambda{\sum\limits_{t = 1}^{P}\left( {{H\left( {{G_{l}\left( {\overset{->}{u}(t)} \right)} - {G\left( {\overset{->}{u}(t)} \right)}} \right)} + {H\left( {{G\left( {\overset{->}{u}(t)} \right)} - {G_{h}\left( {\overset{->}{u}(t)} \right)}} \right)}} \right)}}}} & (59)\end{matrix}$where H(·) represents a non-negative penalty function for violating theconstraints and λ is a user-specified parameter for weighting thepenalty. For large values of λ, the resulting model will observe theconstraints upon the gain. In addition, extra data points which areutilized only in the second part of the cost function may be added tothe historical data set to effectively fill voids in the input space. Byadding these additional points, proper gain extrapolation orinterpolation can be guaranteed. In the preferred embodiment, the gainconstraints are held constant over the entire input space.

By modifying the optimization problem with the gain constraints, modelsthat observe gain constraints can be effectively trained. Byguaranteeing the proper gain, users will have greater confidence that anoptimization and control system based upon such a model will workproperly under all conditions.

One prior art example of guaranteeing global positive or negative gain(monotonicity) in a neural network is described in J. Sill & Y. S.Abu-Mostafa, “Monotonicity Hints,” Neural Information ProcessingSystems, 1996. The technique disclosed in this reference relies onadding an additional term to the cost function. However, this approachcan only be utilized to bound the gain to be globally positive ornegative and is not used to globally bound the range of the gain, nor isit utilized to locally bound the gain (depending on the value of u(t)).

To further elaborate on the problem addressed with gain constrainttraining, consider the following. Neural networks and other empiricalmodeling techniques are generally intended for situations where nophysical (first-principles) models are available. However, in manyinstances, partial knowledge about physical models may be known, asdescribed hereinabove

Because empirical and physical modeling methods have largelycomplementary sets of advantages and disadvantages (e.g., physicalmodels generally are more difficult to construct but extrapolate betterthan neural network models), the ability to incorporate available apriori knowledge into neural network models can capture advantages fromboth modeling methods.

The nature of partial prior knowledge varies by application. Forexample, in the continuous process industries such as refining orchemical manufacturing, operators and process engineers usually know,from physical understanding or experience, a great deal about theirprocesses, including: (1) input-output casualty relations; thisknowledge is easily modeled via network connectivity, (2) input-outputnonlinear/linear relations; this knowledge is easily modeled via networkarchitecture, and (3) approximate bounds on some or all of the gains.This knowledge can be modeled by adding constraints of the form

$\begin{matrix}{\min_{ki}{\leq \frac{\partial y_{k}}{\partial x_{i}} \leq \max_{ki}}} & (60)\end{matrix}$to the training algorithm, where y_(k) is the k^(th) output, x_(i) isthe i^(th) input, and min_(ki) and max_(ki) are constants. Note that anon-zero max_(ki)-min_(ki) range allows for nonlinear input-outputrelationships. By using gain-constrained models, superior performance isachieved both when calculating predictions and when calculating gains.

When calculating predictions, extrapolation (generalization) accuracycan be improved by augmenting gain-constrained training withextrapolation training, in which the gain constraints alone are imposedoutside of the regions populated by training data.

When calculating gains, accuracy is necessary for optimization, control,and other uses. Correlated inputs (substantial cross-correlations amongthe time series of the input variables), and “weak” (inaccurate,incomplete, or closed-loop) data are both very common in realapplications, and frequently cause incorrect gains in neural networkmodels. When inputs are correlated, many different sets of gains canyield equivalent data-fitting quality (the gain solution isunder-specified). Constraining the gains breaks this symmetry byspecifying which set of gains is known a priori to be correct (the gainsolution is fully-specified). Such gain constraints are consistent bydefinition with the data, and therefore do not affect the data-fittingquality. In the case of “weak” data, on the other hand, gain constraintsare imposed to purposefully override, or contradict, the data. A simpleprocess industries example of this case is illustrated in FIG. 27 d, anda simple generic example is given in (Sill and Abu-Mostafa 1977). Suchgain constraints are not consistent with the data, and thereforetypically increase the data-fitting error.

In FIG. 27 d, dots represent scatterplot data of pH versus acid for astirred-tank reactor, where the flowrate of base into the tank isconstant, and the pH is closed-loop controlled to a constant setpointvalue by adjusting the flowrate of acid into the tank. (The plotrepresents a small, linear region of operation.) Whenever the pH rises,the controller increases the acid inflow in order to lower the pH, andvice versa, creating data with a positive pH-acid correlation, as shown,i.e., the data pH-to-acid gain is positive. The correct physicalrelationship, given by the solid line, has a negative pH-to-acid gain,which completely contradicts the data.

Merging prior knowledge with neural networks has been addressed beforein the literature, (Sill and Abu-Mostafa 1997), which provides a broadframework focusing on input-output relations rather than gains. Allprevious work formulates the problem in terms of equality bounds (pointgain targets), although the formulation of (Lampinen and Selonen 1995)can in effect achieve interval targets by means of an associatedcertainty coefficient function.

FIG. 27 e illustrates a screen that allows the modeler to specify, foreach pair of input-output variables, gain constraints (min and maxbounds), and a pair of tolerance parameters. (In addition, the ratio ofextrapolation datapoints (patterns) per training datapoints (patterns)is specifiable.)

As illustrated in FIG. 27 f, the modeler can select a (or any number of)input-output cells, and become able to specify the “Gain Constraints”and “Tolerance (optional)*” values for the cell(s) selected. The MinGain and Max Gain are the bounds within which the modeler wishes thegain to lie when the model has completed training. It is allowable forthe Min Gain value to take its (default) value of −Inf (−infinity),and/or the Max Gain value to take its (default) value of +Inf(+infinity). This is common when the intended bounds are simply positiveor negative—that gain should be of a given sign. The Tolerance Gainparameter extends the specified (Min Gain, Max Gain) interval by thespecified percentage for the purposes of term balancing only. While thegain constraints have priority over the data in the case that they arein conflict, both Tolerance parameters give the modeler a way to affecthow strongly the constraints override the data in the case of conflict(lower values mean strong overriding). The Tolerances allow the model tonot be distorted by attempting to bring the gain of every singledatapoint strictly within the specified range; datapoints that representgain “outliers” can badly distort a model or even prevent it fromconverging if undue and unnecessary attention is given to it. It isusually not necessary to change the Tolerance parameters from theirdefault values of 20%; the default values perfectly achieve the desiredresults in almost all cases.

FIG. 27 g illustrates the Gain Constraints Monitor viewable during andafter training a gain-constrained model. As shown, the values in theMonitor indicate the percentage of the training datapoints (patterns)that violate the (Min Gain, Max Gain) interval; in the case shown, allvalues are well below the 20% Tolerance Pattern parameter value actuallyused to train the model. This information is analogous to thedata-fitting error (Rel Error and R-squared, as shown in the “Train”screen also in FIG. 27 g), but for gain constraints instead of data. Itindicates the degree of conformance to “fitting the gains.” As with poordata-fitting results, any number of remedies are available to themodeler if poor gain constraint conformance occurs; the methods employeddepend heavily on the particulars of the data and the process beingmodeled.

The Objective Function

A neural network with outputs y_(k) and inputs x_(i) is trained subjectto the set of constraints in equation 60. The objective function to beminimized in training isE=E ^(D) +E ^(C)  (61)where E^(D) is the data-fitting error, and E^(C) is a penalty forviolating the constraints. Weighting factors for E^(D) and E^(C) areintroduced later in the form of learning rates (section 2.4). We use thestandard sum of squares for E^(D)

$\begin{matrix}{E^{D} = {{\sum\limits_{k}^{N_{out}}E_{k}^{D}} = {\sum\limits_{p}^{N_{trp}}{\sum\limits_{k}^{N_{out}}\left( {t_{k}^{p} - y_{k}^{p}} \right)^{2}}}}} & (62)\end{matrix}$where p is the pattern index t_(k) ^(p) is the target for output y_(k)^(p), N_(out) is the number of output units, and N_(trp) is the numberof training patterns. Although we assume equation 62 throughout, thealgorithm has no inherent restrictions to this form for E^(D).

E^(C) is written as:

$\begin{matrix}{E^{C} = {{\sum\limits_{k}^{N_{out}}{\sum\limits_{i}^{N_{inp}}E_{ki}^{C}}} = {\sum\limits_{p}^{N_{trp}}{\sum\limits_{k}^{N_{out}}{\sum\limits_{i}^{N_{inp}}{f\left( \frac{\partial y_{k}^{p}}{\partial x_{i}^{p}} \right)}}}}}} & (63)\end{matrix}$where N_(inp) is the number of input units and ƒ measures the degree towhich ∂y_(k) ^(p)/∂x_(i) ^(p) violates its constraints, this illustratedin FIG. 27 h.The Penalty Function.For discussion purposes, n the following, the pattern index p issuppressed for clarity.The penalty function ƒ is defined as follows:

$\begin{matrix}{{{f\left( \frac{\partial y_{k}}{\partial x_{i}} \right)} \equiv f_{ki}} = \begin{Bmatrix}{\frac{\partial y_{k}}{\partial x_{i}} - \max_{ki}} & {{{if}\mspace{14mu}\frac{\partial y_{k}}{\partial x_{i}}} > \max_{ki}} & \left( {\max\mspace{14mu}{is}\mspace{14mu}{violated}} \right) \\{\min_{ki}{- \frac{\partial y_{k}}{\partial x_{i}}}} & {{{if}\mspace{14mu}\frac{\partial y_{k}}{\partial x_{i}}} < \min_{ki}} & \left( {\min\mspace{14mu}{is}\mspace{14mu}{violated}} \right) \\0 & {else} & \left( {{no}\mspace{14mu}{violation}} \right)\end{Bmatrix}} & (64)\end{matrix}$and its derivative as:

$\begin{matrix}{f_{ki}^{\prime} = \begin{Bmatrix}{+ 1} & {{{if}\mspace{14mu}\frac{\partial y_{k}}{\partial x_{i}}} > \max_{ki}} & \left( {\max\mspace{14mu}{is}\mspace{14mu}{violated}} \right) \\{- 1} & {{{if}\mspace{14mu}\frac{\partial y_{k}}{\partial x_{i}}} < \min_{ki}} & \left( {\min\mspace{14mu}{is}\mspace{14mu}{violated}} \right) \\0 & {else} & \left( {{no}\mspace{14mu}{violation}} \right)\end{Bmatrix}} & (65)\end{matrix}$

The function ƒ is shown in FIG. 27 h (such shapes are widely used inpenalty functions). No improvement in solution quality resulted fromusing an everywhere smooth (and more time consuming) approximation ofequation 64.

Gradient Descent Training

From equations 62, 63, 65, and the chain rule, the gradient descentupdate equation for a single pattern is given by:

$\begin{matrix}{{\Delta\; w} = {{- {\sum\limits_{k}^{N_{out}}{\eta_{k}\frac{\partial}{\partial w}\left( {t_{k} - y_{k}} \right)^{2}}}} - {\sum\limits_{k}^{N_{out}}{\sum\limits_{i}^{N_{inp}}{\lambda_{ki}f_{ki}^{\prime}\frac{\partial}{\partial w}\left( \frac{\partial y_{k}}{\partial x_{i}} \right)}}}}} & (66)\end{matrix}$where w denotes and arbitrary weight or bias, η_(k) is the learning ratefor E_(k) ^(D), and λ_(ki) is the learning rate for E_(ki) ^(D).Expressions for the second derivatives ∂/∂w(∂y_(k)/∂x_(i)) are given inthe Appendix. The weights and biases are updated after M patterns (M aparameter) according to:w(T)=Δw(T)+μΔw(T−1)  (67)where T is the weight update index and μ is the momentum coefficient.

An alternative to this gradient descent formulation would be to trainusing a constrained nonlinear programming (NLP) method (e.g. SQP or GRG)instead because (1) commercial NLP codes use Hessian information (eitherexact or approximate), and experience shows that any such explicitlyhigher-order training method tends towards overfitting in a way that isdifficult to control in a problem-independent manner, and (2) themethods described hereinbelow constitute “changes to the problem” forNLPs, which would interfere with their Hessian calculation procedures.

Extrapolation Training

Some degree of improved model extrapolation may be expected fromgain-constrained training solely from enforcing correct gains at theboundaries of the training data set. However, extrapolation trainingexplicitly improves extrapolation accuracy by shaping the responsesurface outside the training regions. Extrapolation training applies thegain constraint penalty terms alone to the network for patternsartificially generated anywhere in the entire input space, whether ornot training patterns exist there. An extrapolation pattern consists ofan input vector only—no output target is needed, as the penalty terms donot involve output target values. The number of extrapolation patternsto be included in each training epoch is an adjustable parameter.Extrapolation input vectors are generated at random from a uniformdistribution covering all areas of the input space over which theconstraints are assumed to be valid.

Training Procedure Summary

The training algorithm proceeds as follows:

-   -   1. Backpropagation is performed on E^(D), and the Δw's in        equation 66 are incremented with the first term contributions.    -   2. A backward pass for each y^(k) is performed to compute        ∂_(yk)/∂x_(i). This is done by clearing the propagation error        δ's for all units, setting δ_(yk)=y_(k)′, and backpropagating to        the inputs, which yields δx_(i)=∂_(yk)/∂x_(i). This follows        easily from the definition of δ and this technique has long been        known to practitioners.    -   3. Quantities computed immediate preceding step 2 are used to        evaluate the ∂/∂w(∂y_(k)/∂x_(i) expressions given in the        Appendix, and the Δw's in equation 66 are incremented with the        second term contributions.    -   4. The weights are updated after every M patterns (1≦M≦?)        according to equation 67.

For an extrapolation pattern, step 1 is omitted and step 2 is precededby a forward pass.

Dynamic Balancing of Objective Function Terms

If the constraints do not conflict with the data, balancing the relativestrengths (learning rates) of the data and penalty terms in theobjective function is relatively straightforward. If the constraintsconflict with the data, however, a robust method for term balancing maybe required.

While the constraints have priority over data in case of conflict,insisting that all constraint violations be zero is ordinarily neithernecessary not conducive to obtaining good models (models of practicalvalue) in real applications. Typically, it is not desirable to drive‘constant outliers” of negligible practical consequence into absoluteconformance at the expense of significantly worsening the data fit.

Instead, the method of the present disclosure allows, for eachconstraint, a specified fraction of training patterns (typically 10-20%)to violate the constraint to any degree, before deeming the constraintto be in violation. The violating patterns may be different for eachconstraint, and the violating patterns for each constraint may be chosenfrom the entire training set. The resulting combinatorics allows thenetwork enormous degrees of freedom to achieve maximum data-fittingwhile satisfying the constraints.

Dynamic term balancing proceeds as follows. The initial values of theη_(k) are set to a standard value. For each output, the initial valuesfor the λ_(ki) are computed before training begins such that the averagefirst level derivatives of the data term (for that output) isapproximately equal to the first level derivative of the sum of theconstraint terms (for that output). The learning rates η_(k) and λ_(ki)and then adjusted after each epoch according to the current state of thetraining process.

A wide variety of situations are possible in gain constrained training,such as: constraints may be consistent or inconsistent with the data;constraints may be violated at the outset of training or only much laterafter the gains have grown in magnitude; the constraints may bemis-specified and cannot be achieved at all; and so on. Hence, a varietyof rules for adjusting the learning rates are necessary.

To adjust η_(k) and λ_(ki) after each epoch, we use an extension of therapid/slow η_(k) dynamics described in for training backprop nets, whereη_(k) is slowly (additively) increased if the error is decreasing and israpidly (multiplicatively) decreased if the error is increasing. In thefollowing, a “large” change indicates a multiplicative change and a“small” change indicates an additive change.

The fundamental goal that the constraints should override the data incase of conflict dictates the following basic scheme. (λ_(ki) is ofcourse subject to change only if a constraint for input-output variablepair ki has been specified).

λ_(ki):

(a) large increase if constraint ki is not satisfied (“satisfaction” isdefined above in this section).

-   (b) small decrease if constraint _(ki) is satisfied.    η_(k):-   (a) large decrease if E^(D) has worsened (increased).-   (b) small decrease if E^(D) has improved (decreased).

Some exceptions to these rules are required, however, which arepresented in correspondence with the above rules.

λ_(ki):

-   (a) no increase is made if E_(ki) ^(C) improved, since, given that    E_(ki) ^(C) is improving, increasing ki would make it unnecessarily    harder for E^(D) to improve.-   (b) no decrease is made if all constraints are exactly satisfied    (E^(C)=0). In this case, either (a) the constraints agree with the    data, in which case retaining the λ_(ki) level will actually speed    convergence, or (b) the constraints are such that they require large    gains to be violated, and will therefore remain satisfied until    later in the training process when the weights have grown    sufficiently to produce large gains. In this case, repeatedly    decreasing the λ_(ki)'s early in the training process would result    in their being too small to be effective when such constraints    become violated.    η_(k):-   (a) no decrease is made in two cases: (1) all constraints are    satisfied but E^(C) has increased. Both E^(D) and E^(C) are    increasing, but the decrease in λ_(ki) (due to the constraints being    satisfied) might allow E^(D) to improve without decreasing    η_(k). (2) all the constraints are not satisfied, but E^(C) has    improved; this improvement may allow E^(D) to improve without    increasing η_(k).-   (b) no increase is made if E^(C) has increased, in accordance with    allowing the constraints to dominate.

Parameter values used in these dynamics depend on the networkarchitecture, but for typical networks are generally in the range of1.3*λ_(ki) and 0.8*η_(k) for multiplicative changes, and (−)0.03*λ_(ki)and (+)0.02*η_(k) for additive changes. Also, expected statisticalfluctuations in training errors are accounted for when deeming whetheror not E^(D), E^(C) or E_(ki) ^(C) have increased.

Left and Right Gain Constraints

However, a limitation with the above described gain constraint method isthat the gain constraints must be specified with the same min_(ki) andmax_(ki) constants, for all input data points. In a further embodiment,the method would be to allow the user to specify different gainconstraint bounds depending on where the input lies in the input space.However, when there are even just a few input variables, specifying manyvalues of constraint bounds for each of the input variables becomescombinatorially explosive.

For a simple example, the user is unable with the above described methodto specify, for instance, one set of gain bounds at the minimum value ofx₁ and another set of gain bounds at the maximum value of x₁. If theuser were able to specify this information, the user could, forinstance, specify a positive gain at the minimum value of x₁, and anegative gain at the maximum value of x₁.

Given such a specification, the network could be visualized as creatingan input-output relationship that would resemble a parabola with thepeak pointing upwards. However, because a neural network has manydegrees of freedom, the input-output relationship formed may contain afew to several “humps” (changes of the sign of the gain) between theminimum and the maximum of x₁.

In modeling the process industries, it has been observed that a singlechange in the sign of an input-output gain suffices to model theoverwhelming majority of processes. To accommodate for this with gainconstraints, a simplified model is provided that limits the number ofchanges in the sign of the gain to one change. In conjunction with thismodel, we allow the specification of “left” (minimum) and “right”(maximum) gain bounds.

The simplified model, which limits the number changes in the sign of thegain to one change, is given as follows:

-   -   x_(i) denotes an arbitrary input variable, indexed by i. y        denotes the output of the model (multiple outputs are treated as        separate models).

The model is given by:

$\begin{matrix}{y = {b + {\sum\limits_{i}^{N}\left( {{m_{i}x_{i}} + {g_{i}{f\left( u_{i} \right)}}} \right)}}} & (68)\end{matrix}$

-   -   where N is the number of inputs x_(i), u_(i) is an intermediate        “convenience” variable, given by

$\begin{matrix}{u_{i} = \frac{x_{i} - x_{i}^{a} - {\frac{1}{2}\left( {x_{i}^{b} - x_{i}^{a}} \right)}}{\frac{1}{4}\left( {x_{i}^{b} - x_{i}^{a}} \right)}} & (69)\end{matrix}$

the function f(u_(i)) is given by

$\begin{matrix}{f = {\ln\frac{1}{1 + {\mathbb{e}}^{u_{i}}}}} & (70)\end{matrix}$

and the regression parameters are b, x_(i) ^(a), x_(i) ^(b), m_(i), andg_(i)

The function y is a model which has at most one change in the sign of∂y/∂_(i), for any x_(i) (at most one “bend” in the function), and can bestraight (no bends). The bends occur at x_(a) (on the left) and at x_(b)(on the right). The “ends” of the function have constant slopes at plusand minus infinity.

As described hereinabove, because neural networks have many availableparameters, overfitting (overtraining) is an issue, and using NLP's(optimizers) to perform the regression optimization was avoided. Theother reason NLP's were avoided was to allow extrapolation training.With the above simplified model, on the other hand, neither of theseissues are relevant—overfitting cannot occur due to the limited numberof parameters in the model, and extrapolation training is not an issuesince the gains of the model may be specified by the user at the minimumand maximum values of each input, thereby specifying what shape themodel should take in all regions of input space.

Therefore, the method of choice is to use an NLP to perform theconstrained regression in the case of this simplified model. Theobjective function is in this case simply given by the Equation:

$\begin{matrix}{E = {{\sum\limits_{k}^{N_{out}}E_{k}} = {\sum\limits_{p}^{N_{trp}}{\sum\limits_{k}^{N_{out}}\left( {t_{k}^{p} - y_{k}^{p}} \right)^{2}}}}} & (71)\end{matrix}$where p is the data index, t_(k) ^(p) is the measures “target” value foroutput y_(k) ^(p), N_(out) is the number of output units, and N_(trp) isthe number of data points in the set of data used to perform theregression, typically called the “training se.”

The constraints in the NLP are the two gain bounds given for each inputvariable, one at the minimum value and one at the maximum value of eachx_(i).

The gain expression for the gain of y with respect to a given inputx_(i) is given by

$\begin{matrix}{\frac{\partial y}{\partial x_{i}} = {\frac{1}{\frac{1}{4}\left( {x_{i}^{b} - x_{i}^{a}} \right)}\left( {m_{i} - {g_{i}\frac{1}{1 + {\mathbb{e}}^{u_{i}}}}} \right)}} & (72)\end{matrix}$where, again, μ_(i) is given in terms of xiby equation 69.

Thus, the user specifies as constraints to the NLP, bounds for∂x/∂y_(i), at both the minimum and maximum values of each x_(i). Thederivative is evaluated in each case given the minimum or maximum valueof the x_(i).

Adding Interactions to the Left and Right Gain Constraints

A notable feature of the above model given by equation 68, is that,unlike the neural network model, there are no “interactions”possible—that is, no product of inputs terms possible in the model, suchas x₁*x₂. In that process industries, it has been observed that thislimitation is of minimal nuisance. However, it is easy to augment thisLeft and Right Gain Constraints Method to allow specific interactionswhich the user knows via process knowledge to be of value to the model.For example, then the gain_(x1)=f(x₂) and gain_(x2)=f(x₁) for the termrelationship x₁*x₂, such that there is a relationship of the gain of onevariable to another.

The user may specify an interaction of any term (non-additive, asadditive “interactions” are already present) by simple adding a ‘dummyvariable,” say x_(n+1) to the model, and setting it equal to the desiredinteraction of the variables of interest as a constraint to the NLP.That is, for example, the model would be augmented as follows:

$\begin{matrix}{y = {b + {\sum\limits_{i}^{N + 1}\left( {{m_{i}x_{i}} + {g_{i}{f\left( u_{i} \right)}}} \right)}}} & (73)\end{matrix}$Here, a constraint would be added to the NLP specifying thatx_(N+1)=x_(p)x_(q)  (74)for example. The intermediate variable u_(n+1) would then be computed as

$\begin{matrix}{u_{N + 1} = \frac{\left( {x_{p}x_{q}} \right) - x_{N + 1}^{a} - {\frac{1}{2}\left( {x_{N + 1}^{b} - x_{N + 1}^{a}} \right)}}{\frac{1}{4}\left( {x_{N + 1}^{b} - x_{N + 1}^{a}} \right)}} & (75)\end{matrix}$Identification of SS Model with Dynamic Data

Referring now to FIG. 28, there is illustrated a block diagram of aprior art Weiner model, described in M. A. Henson and D. F. Seborg,“Nonlinear Process Control,” Prentice Hall PTR, 1997, Chapter 2, pp11-110. In the Weiner model, a non-linear model 376 is generated. Thisnon-linear model is a steady-state model. This steady-state model may betrained on input data u(t) to provide the function y(t)=f(u(t)) suchthat this is a general non-linear model. However, the input u(t) isprocessed through a linear dynamic model 374 of the system, which lineardynamic model 374 has associated therewith the dynamics of the system.This provides on the output thereof a filtered output ū(t) which has thedynamics of the system impressed thereupon. This constitutes the inputto the non-linear model 376 to provide on the output a prediction y(t).

Referring now to FIG. 29, there is illustrated a block diagram of thetraining method of the present embodiment. A plant 378 is provided whichcan represent any type of system to be modeled, such as a boiler or achemical process. There are various inputs provided to the plant in theform of u(t). This will provide an actual output y^(a)(t). Although notillustrated, the plant has a number of measurable state variables whichconstitute the output of various sensors such as flow meters,temperature sensors, etc. These can provide data that is utilized forvarious training operations, these state outputs not illustrated, itbeing understood that the outputs from these devices can be a part ofthe input training data set.

A steady-state neural network 379 is provided which is a non-linearnetwork that is trained to represent the plant. A neural networktypically contains an input layer and an output layer and one or morehidden layers. The hidden layers provide the mapping for the inputlayers to the output layers and provide storage for the storedrepresentation of the plant. As noted hereinabove, with a sufficientamount of steady-state data, an accurate steady-state model can beobtained. However, in a situation wherein there is very littlesteady-state data available, the accuracy of a steady-state model withconventional training techniques is questionable. As will be describedin more detail hereinbelow, the training method of the presentembodiment allows training of the neural network 374, or any otherempirical modeling method to learn the steady-state process model fromdata that has no steady-state behavior, i.e., there is a significantdynamic component to all training data.

Typically, a plant during operation thereof will generate historicaldata. This historical data is collected and utilized to later train anetwork. If there is little steady-state behavior exhibited in the inputdata, the present embodiment allows for training of the steady-statemodel. The input data u(t) is input to a filter 381 which is operable toimpress upon the input data the dynamics of the plant 378 and thetraining data set. This provides a filtered output u^(f)(t) which isinput to a switch 380 for input to the plant 378. The switch 380 isoperable to input the unfiltered input data u(t) during operation of theplant, or the filtered input data u^(f)(t) during training into theneural network 379. As will be described hereinbelow, the u(t) inputdata, prior to being filtered, is generated as a separate set of dynamictraining data by a step process which comprises collecting step data ina local region. The filter 381 has associated therewith a set of systemdynamics in a block 382 which allows the filter 381 to impress thedynamics of the system onto the input training data set. Therefore,during training of the neural network 379, the filtered data ū(t) isutilized to train the network such that the neural network 379 providesan output y(t) which is a function of the filtered data ū(t) or:y ^(p)(t)=f({right arrow over (u)} ^(f)(t))  (76)

Referring now to FIG. 30, there is illustrated a diagrammatic view ofthe training data and the output data. The training data is the actualset of training data which comprises the historical data. This is theu(t) data which can be seen to vary from point to point. The problemwith some input data in a training set of data, if not all data, is thatit changes from one point to another and, before the system has“settled,” it will change again. That is, the average time betweenmovements in u(t) is smaller than T_(ss), where T_(ss) is the time fory(t) to reach steady-state. As such, the corresponding output data y(t)will constitute dynamic data or will have a large dynamic componentassociated therewith. In general, the presence of this dynamicinformation in the output data must be accounted for to successfullyremove the dynamic component of the data and retain the steady-statecomponent of the steady-state neural network 379.

As will be described in more detail hereinbelow, the present embodimentutilizes a technique whereby the actual dynamics of the system which areinherent in the output data y(t) are impressed upon the input data u(t)to provide filtered input data ū(t). This data is scaled to have a gainof one, and the steady-state model is then trained upon this filtereddata. As will also be described in more detail hereinbelow, the use ofthis filtered data essentially removes the dynamic component from thedata with only the steady-state component remaining. Therefore, asteady-state model can be generated.

Referring now to FIG. 31, there is illustrated a flow chart depictingthe training procedure for the neural network 379 of FIG. 29 for asingle output. As noted above, the neural network is a conventionalneural network comprised of an input layer for receiving a plurality ofinput vectors, an output layer for providing select predicted outputs,and one or more hidden layers which are operable to map the input layerto the output layer through a stored representation of the plant 378.This is a non-linear network and it is trained utilizing a training dataset of input values and target output values. This is, as describedhereinabove, an iterative procedure utilizing algorithms such as thebackpropagation training technique. Typically, an input value is inputto the network during the training procedure, and also target data isprovided on the output. The results of processing the input data throughthe network are compared to the target data, and then an errorgenerated. This error, with the backpropagation technique, is then backpropagated through the network from the output to the input to adjustthe weights therein, and then the input data then again processedthrough the network and the output compared with the target data togenerate a new error and then the algorithm readjusts the weights untilthey are reduced to an acceptable level. This can then be used for allof the training data with multiple passes required to minimize the errorto an acceptable level, resulting in a trained network that providestherein a stored representation of the plant.

As noted above, one of the disadvantages to conventional trainingmethods is that the network 379 is trained on the set of historicalinput data that can be incomplete, or have some error associatedtherewith. The incompleteness of the historical data may result in areasin the input space on which the network is not trained. The network,however, will extrapolate its training data set during the trainingoperation, and actually provide a stored representation within thatportion of the input space in which data did not exist. As such,whenever input data is input to the network in an area of the inputspace in which historical input data did not exist during training, thenetwork will provide a predicted output value. This, however,effectively decreases the confidence level in the result in this region.Of course, whenever input data is input to the network in a region thatwas heavily populated with input data, the confidence level isrelatively high.

Another source of error, as noted hereinabove, is the dynamic componentof the data. If the historical data that forms the training data set isdynamic in nature, i.e., it is changing in such a manner that the outputnever settles down to a steady-state value, this can create some errorswhen utilizing this data for training. The reason for this is that thefundamental assumption in training a steady-state neural network with aninput training data set is that the data is steady-state data. Thetraining procedure of the present embodiment removes this error.

Referring further to FIG. 31, the flow chart is initiated at a block 384and then proceeds to a block 386 wherein dynamic data for the system iscollected. In the preferred embodiment, this is in the form of step testdata wherein the input is stepped between an initial value and a finalvalue multiple times and output data taken from the plant under theseconditions. This output data will be rich in dynamic content for a localregion of the input space. An alternative method is to examine thehistorical data taken during the operation of the plant and examine thedata for movements in the manipulated variables (MVs) or the dynamicvariables (DVs). These variables are then utilized for the purpose ofidentifying the dynamic model. However, the preferred model is toutilize a known input that will result in the dynamic change in theoutput. Of course, if there are no dynamics present in the output, thenthis will merely appear as a steady-state value, and the dynamic modelwill have filter values of a=0 and b=0. This will be describedhereinbelow.

The step test data, as will be described hereinbelow, is data that istaken about a relatively small region of the input space. This is due tothe fact that the variables are only manipulated between two values, andinitial steady-state value and a final value, in a certain region of theinput space, and the data is not taken over many areas of the inputspace. Therefore, any training set generated will represent only a smallportion of the input space. This will be described in more detailhereinbelow. It should be noted that these dynamics in this relativelysmall region of the input space will be utilized to represent thedynamics over the entire input space. A fundamental presumption is thatthe dynamics at any given region remain substantially constant over theentire input space with the exception of the dynamic gain varying.

Once the dynamic data has been collected for the purpose of training,this dynamic training data set is utilized to identify the dynamic modelof the system. If, of course, a complete steady-state data set wereavailable, there would be a reduced need for the present embodiment,although it could be utilized for the purpose of identifying thedynamics of the system. The flow chart then proceeds to a block 387wherein the dynamics of the plant are identified. In essence, aconventional model identification technique is utilized which models thedynamics of the plant. This is a linear model which is defined by thefollowing equation:y(t)=−a ₁ y(t−1)−a ₂(t−2)+b ₁ u(t)+b ₂ u(t−1)  (77)In the above-noted model of Equation 77, the values of a₁, a₂, b₁ and b₂define the parameters of the model and are defined by training thismodel. This operation will be described in detail hereinbelow; however,once trained, this model will define the dynamic model of the plant 378as defined by the dynamics associated with the dynamic training data setat the location in the input space at which the data was taken. Thiswill, of course, have associated therewith a dynamic gain, which dynamicgain will change at different areas in the input space.

Once the dynamic model has been identified utilizing the dynamictraining data set, i.e., the a's and b's of the model have beendetermined, the program will flow to a function block 388 to determinethe properties of a dynamic pre-filter model, which is operable toprocess the input values u(t) through the dynamic model to provide afiltered output u^(f)(t) on the output which is, in effect, referred toas a “filtered” input in accordance with the following equation:{right arrow over (u)} ^(f)(t)=a ₁ {right arrow over (u)}(t−1)−a ₂{right arrow over (u)}(t−1)+ b ₁ u(t)+ b ₂ u(t−1)  (78)wherein the values of a₁ and a₂ are the same as in the dynamic model ofthe plant, and the values of b₁ and b₂ are adjusted to set the gain to avalue of zero.

The pre-filter operation is scaled such that the gain of the dynamicmodel utilized for the pre-filter operation is set equal to unity. Theb-values are adjusted to provide this gain scaling operation. The gainis scaled in accordance with the following:

$\begin{matrix}{{gain} = {1 = \frac{{\overset{\_}{b}}_{1} + {\overset{\_}{b}}_{2}}{1 + a_{1} + a_{2}}}} & (79)\end{matrix}$If the gain were not scaled, this would require some adjustment to thesteady-state model after training of the steady-state model. Forexample, if the gain of the model were equal to “two,” this wouldrequire that the steady-state model have a gain adjustment of “one-half”after training.

After the filter values have been determined, i.e., the b-values withthe gain set equal to one, then the input values u(t) for the historicaldata are processed through the pre-filter with the gain set equal to oneto yield the value of ū(t), as indicated by a function block 390. Atthis point, the dynamics of the system are now impressed upon thehistorical input data set, i.e., the steady-state component has beenremoved from the input values. These input values ū(t) are now input tothe neural network in a training operation wherein the neural network378 is trained upon the filtered input values over the entire inputspace (or whatever portion is covered by the historical data). This dataū(t) has the dynamics of the system impressed thereupon, as indicated byblock 391. The significance of this is that the dynamics of the systemhave now been impressed upon the historical input data and thus removedfrom the output such that the only thing remaining is the steady-statecomponent. Therefore, when the neural network 378 is trained on thefiltered output, the steady-state values are all that remain, and avalid steady-state model is achieved for the neural network 378. Thissteady-state neural network is achieved utilizing data that has verylittle steady-state nature. Once trained, the weights of the neuralnetwork are then fixed, as indicated by a function block 392, and thenthe program proceeds to an END block 394.

Referring now to FIG. 32, there is illustrated a diagrammatic view ofthe step test. The input values u(t) are subjected to a step responsesuch that they go from an initial steady-state value u_(i)(t) to a finalvalue u_(f)(t). This results in a response on the output y(t), which isrich in dynamic content. The step test is performed such that the valueof u(t) is increased from u_(i)(t) to u_(f)(t), and then decreased backto u_(i)(t), preferably before the steady-state value has been reached.The dynamic model can then be identified utilizing the values of u(t)presented to the system in the step test and the output values of y(t),these representing the dynamic training data set. This information isutilized to identify the model, and then the model utilized to obtainthe pre-filtered values of ū(t) by passing the historical input datau(t) through the identified model with the dynamic gain of the model setequal to one. Again, as noted above, the values of b are adjusted in themodel such that the gain is set equal to one.

Referring now to FIG. 33, there is illustrated a diagrammatic view ofthe relationship between the u(t) and the ū(t), indicating that the gainis set equal to one. By setting the gain equal to one, then only thedynamics determined at the “training region” will be impressed upon thehistorical input data which exists over the entire input space. If theassumption is true that the only difference between the dynamics betweengiven regions and the input space is the dynamic gain, then by settingthe gain equal to one, the dynamics at the given region will be true forevery other region in the input space.

Referring now to FIG. 34, there is illustrated a block diagram of thesystem for training a given output. The training system for theembodiment described herein with respect to impressing the dynamics ofthe plant onto the historical input data basically operates on a singleoutput. Most networks have multiple inputs and multiple outputs and arereferred to as MIMO (multi-input, multi-output) networks. However, eachoutput will have specific dynamics that are a function of the inputs.Therefore, each output must have a specific dynamic model which definesthe dynamics of that output as a function of the input data. Therefore,for a given neural network 400, a unique pre-filter 402, will berequired which receives the input data on the plurality of input lines404. This pre-filter is operable to incorporate a dynamic model of theoutput y(t) on the input u(t). This will be defined as the function:

This represents the dynamic relationship between the inputs and a singleoutput with the gain set equal to unity.y(t)=f _(d) ^(local)(ū(t))  (80)

Referring now to FIG. 35, there is illustrated a dynamic representationof a MIMO network being modeled. In a MIMO network, there will berequired a plurality of steady-state neural networks 410, labeled NN₁,NN₂, . . . NN_(M). Each one is associated with a separate output y₁(t),y₂(t), . . . y_(M)(t). For each of the neural networks 410, there willbe a pre-filter or dynamic model 412 labeled Dyn₁, Dyn₂, . . . Dyn_(M).Each of these models 412 receives on the input thereof the input valuesu(t), which constitutes all of the inputs u₁(t), u₂(t), . . . u_(n)(t).For each of the neural networks 410, during the training operation,there will also be provided the dynamic relations hip between the outputand the input u(t) in a block 14. This dynamic relationship representsonly the dynamic relationship between the associated one of the outputsy₁(t), y₂(t), . . . y_(M)(t). Therefore, each of these neural networks410 can be trained for the given output.

Referring now to FIG. 36, there is illustrated the block diagram of thepredicted network after training. In this mode, all of the neuralnetworks 410 will now be trained utilizing the above-noted method ofFIG. 30, and they will be combined such that the input vector u(t) willbe input to each of the neural networks 410 with the output of each ofthe neural networks comprising one of the outputs y₁(t), y₂(t), . . .y_(M)(t).

Graphical Interface for Model Identification

Referring now to FIG. 37, there is illustrated a graphical userinterface (GUI) for allowing the user to manipulate data on the screen,which manipulated data then defines the parameters of the modelidentification procedure. In FIG. 39 there are illustrated two inputvalues, one labeled “reflux” and one labeled “steam.” The reflux data isrepresented by a curve 400 whereas the steam data is represented bycurve 402. These curves constitute dynamic step data which would havecorresponding responses on the various outputs (not shown). As notedhereinabove with respect to FIG. 34, the output would have a dynamicresponse as a result of the step response of the input.

The user is presented the input data taken as a result of the step teston the plant and then allowed to identify the model from this data. Theuser is provided a mouse or similar pointing device (not shown) to allowa portion of one or more of the data values to be selected over a userdefined range. In FIG. 37, there is illustrated a box 404 in phantomabout a portion of the input reflux data which is generated by the userwith the pointing device. There is also illustrated a box 406 in phantomabout a portion of the steam data on curve 402. The portion of each ofthe curves 400 and 402 that is enclosed within the respective boxes 404and 406 is illustrated in thick lines as “selected” data. As notedhereinabove, the step test data is taken in a particular localizedportion of the input space, wherein the input space is defined by thevarious input values in the range over which the data extends. Byallowing the user the versatility of selecting which input data is to beutilized for the purpose of identifying the model, the user is nowpermitted the ability to manipulate the input space. Once the user hasselected the data that is to be utilized and the range of data, the userthen selects a graphical button 410 which will then perform an“identify” operation of the dynamic model utilizing the selectedinformation.

Referring now to FIG. 38, there is illustrated a flowchart depicting thegeneral identify operation described above with respect to FIG. 39. Theprogram is initiated at a function block 412 which indicates a givendata set, which data set contains the step test data for each input andeach output. The program then flows to a function block 414 to displaythe step test data and then to a function block 416 to depict theoperation of FIG. 38 wherein the user graphically selects portions ofthe display step test data. The program then flows to a function block418 wherein the data set is modified with the selected step test data.It is necessary to modify the data set prior to performing theidentification operation, as the identification operation utilizes theavailable data set. Therefore, the original data set will be modified toonly utilize the selected data, i.e., to provide a modified data set foridentification purposes. The program will then flow to a function block420 wherein the model will be identified utilizing the modified dataset.

Referring now to FIG. 39, there is illustrated a second type ofgraphical interface. After a model has been created and identified, itis then desirable to implement the model in a control environment tocontrol the plant by generating new input variables to change theoperation of the plant for the purpose of providing a new and desiredoutput. However, prior to placing the model in a “run-time” mode, it maybe desirable to run a simulation on the model prior to actuallyincorporating it into the run time mode. Additionally, it may bedesirable to graphically view the system when running to determine howthe plant is operating in view of what the predicted operation is andhow that operation will go forward in the future.

Referring further to FIG. 40, there is illustrated a plot of a singlemanipulatable variable (MV) labeled “reflux.” The plot illustrates twosections, a historical section 450 and a predictive section 452. Thereis illustrated a current time line 454 which represents the actual valueat that time. The x-axis is comprised of the steps and horizontal axisillustrates the values for the MV. In this example of FIG. 41, thesystem was initially disposed at a value of approximately 70.0. Inhistorical section 450, a bold line 456 illustrates the actual values ofthe system. These actual values can be obtained in two ways. In asimulation mode, use the actual values that are input to the simulationmodel. In the run-time mode, they are the set point values input to theplant.

In a dynamic system, any change in the input will be made in a certainmanner, as described hereinabove. It could be a step response or itcould have a defined trajectory. In the example of FIG. 40, the changein the MV is defined along a desired trajectory 458 wherein the actualvalues are defined along a trajectory 460. The trajectory 458 is definedas the “set point” values. In a plant, the actual trajectory may nottrack the setpoints (desired MVs) due to physical limitations of theinput device, time constraints, etc. In the simulation mode, these arethe same curve. It can be seen that the trajectory 460 continues up tothe current time line 454. After the current time line 454, there isprovided a predicted trajectory which will show how it is expected theplant will act and how the predictive model will predict. It is alsonoteworthy that the first value of the predicted trajectory is theactual input to the plant. The trajectory for the MV is computed everyiteration using the previously described controller. The user thereforehas the ability to view not only the actual response of the MV from ahistorical standpoint but, also the user can determine what the futureprediction will be a number of steps into the future.

A corresponding controlled variable (CV) curve is illustrated in FIG.42. In this figure, there is also a historical section 450 and apredictive section 452. In the output, there is provided therein adesired response 462 which basically is a step response. In general, thesystem is set to vary from an output value of 80.0 to an input value ofapproximately 42.0 which, due to dynamics, cannot be achieved in thereal world. Various constraints are also illustrated with a upper fuzzyconstraint at a line 464 and a lower fuzzy constraint at a line 466. Thesystem will show in the historical section the actual value on a boldline 468 which illustrates the actual response of the plant, this notedabove as being either a simulator or by the plant itself. It should beremembered that the user can actually apply the input MV while the plantis running. At the current time line 454, a predicted value is shownalong a curve 470. This response is what is predicted as a result of theinput varying in accordance with the trajectory of FIG. 41. In additionto the upper and lower fuzzy constraints, there are also provided upperand lower hard constraints and upper and lower frustum values, that weredescribed hereinabove. These are constraints on the trajectory which aredetermined during optimization.

By allowing the user to view not only historical values but futurepredicted values during the operation of a plant or even during thesimulation of a plant, the operator is now provided with information asto how the plant will operate during a desired change. If the operationfalls outside of, for example, the upper and lower frustum of a desiredoperation, it is very easy for the operator to make a change in thedesired value to customize this. This change is relatively easy to makeand is made based upon future predicted behavior as compared tohistorical data.

Referring now to FIG. 41, there is illustrated a block diagram of acontrol system for a plant that incorporates the GUI interface.Basically, there is illustrated a plant 480 which has a distributedcontrol system (DCS) 482 associated therewith for generating the MVs.The plant outputs the control variables (CV). A predictive controller484 is provided which is basically the controller as noted hereinabovewith respect to FIG. 2 utilized in a control environment for predictingthe future values of the manipulated variables MV(t+1) for input to theDCS 482. This will generate the predicted value for the next step. Thepredictive controller requires basically a number of inputs, the MVs,the output CVs and various other parameters for control thereof. A GUIinterface 490 is provided which is operable to receive the MVs, the CVs,the predicted manipulated variables Mv(t+1) for t+1, t+2, t+3, . . .t+n, from the predictive controller 484. It is also operable to receivea desired control variable CV^(D). The GUI interface 490 will alsointerface with a display 492 to display information for the user andallow the user to input information therein through a user input device494, such as a mouse or pointing device. The predictive controller alsoprovides to the GUI interface 490 a predicted trajectory, whichconstitutes at one point thereof MV(t+1). The user input device 494, thedisplay 492 and the GUI interface 490 are generally portions of asoftware program that runs on a PC, as well as the predictive controller484.

In operation, the GUI interface 490 is operable to receive all of theinformation as noted above and to provide parameters on one or morelines 496 to the predictive controller 484 which will basically controlthe predictive controller 484. This can be in the form of varying theupper and lower constraints, the desired value and even possiblyparameters of the identifying model. The model itself could, in thismanner, be modified with the GUI interface 490.

The GUI interface 490 allows the user to identify the model as notedhereinabove with respect to FIG. 39 and also allows the user to view thesystem in either simulation mode or in run time mode, as noted in FIG.41. For the simulation mode, a predictive network 498 is provided whichis operable to receive the values MV(t+1) and output a predicted controlvariable rather than the actual control variable. This is describedhereinabove with reference to FIG. 2 wherein this network is utilized inprimarily a predictive mode and is not utilized in a control mode.However, this predictive network must have the dynamics of the systemmodel defined therein.

Referring now to FIGS. 42-45, there is illustrated a screen view whichcomprises the layout screen for displaying the variables, both input andoutput, during the simulation and/or run-time operation. In the view ofFIG. 42, there is illustrated a set-up screen wherein four variables canbe displayed: reflex, steam, top_comp, and bot_comp. These are displayedin a box 502. On the right side of the screen are displayed four boxes504, 506, 508 and 510, for displaying the four variables. Each of thevariables can be selected for which box they will be associated with.The boxes 504-510 basically comprise the final display screen duringsimulation. The number of rows and columns can be selected with boxes512.

Referring now to FIG. 42, there is illustrated the screen that willresult when the system is “stepped” for the four variables as selectedin FIG. 43. This will result in four screens displaying the two MVs,reflex and steam, and the two CVs, top_comp and bot_comp. In somesituations, there may be a large number of variables that can bedisplayed on a single screen; there could be as many as thirty variablesdisplayed in the simulation mode as a function of time. In theembodiment of FIG. 44, there are illustrated four simulation displays516, 518, 520 and 522, associated with boxes 504-510, respectively.

Referring now to FIG. 43, there is illustrated another view of thescreen of FIG. 42 with only one row selected with the boxes 512 in twocolumns. This will result in two boxes 524 and 528 that will be disposedon the final display. It can be seen that the content of these twoboxes, after being defined by the boxes 512, is defined by moving to thevariable box 502 and pointing to the appropriate one of the variablenames, which will be associated with the display area 524 or 528. Oncethe variables have been associated with the particular display, then theuser can move to the simulation screen illustrated in FIG. 45, whichonly has two boxes 530 and 532 associated with the boxes 524 and 528 inFIG. 44. Therefore, the user can very easily go into the set-up screenof FIGS. 42 and 44 to define which variables will be displayed duringthe simulation process or during run-time execution.

On-Line Optimizer

Referring now to FIG. 46, there is illustrated a block diagram of aplant 600 utilizing an on-line optimizer. This plant is similar to theplant described hereinabove with reference to FIG. 6 in that the plantreceives inputs u(t) which are comprised of two types of variables,manipulatable variables (MV) and disturbance variables (DV). Themanipulatable variables are variables that can be controlled such asflow rates utilizing flow control valves, flame controls, etc. On theother hand the disturbance variables are inputs that are measurable butcannot be controlled such as feed rates received from another system.However, they all constitute part of the input vector u(t). The outputof the plant constitutes the various states that are represented by thevector y(t). These outputs are input to an optimizer 602 which isoperable to receive desired values and associated constraints andgenerate optimized input desired values. Since this is a dynamic system,the output of the optimizer 602 is then input to a controller 604 whichgenerates the dynamic movements of the inputs which are input to thedistributed control system 606 for generation of the inputs to the plant600, it being understood that the DCS 606 will only generate the MVsthat are actually applied to the plant. The distinction of this systemover other systems is that the optimizer 602 operates on-line. Thisaspect is distinctive from previous system in that the dynamics of thesystem must be accounted for during the operation of this optimizer. Thereason for this is that when an input value moves from one value toanother value, there are dynamics associated therewith. These dynamicsmust be considered or there will multiple errors. This is due to thefact that most predictive systems utilizing optimizers implement theoptimization routine with steady state models. No decisions can be madeuntil the plant settles out, such that such an optimization must beperformed off-line.

Referring now to FIG. 47, there is illustrated a block diagram of theoptimizer 602. In general, there is provided a nonlinear steady statemodel 610 which is operable to receive control variables (CV) which arebasically the outputs of the plant y(t). The steady state model 610 isoperable to receive the CV values as an input in the form of set pointsand generates an output optimized value of u(t+1). This represents theoptimized or predicted future value that is to be input to thecontroller for controlling the system. However, the steady state model610 was generated with a training set of input vectors and outputvectors that represented the plant at the time that this training datawas taken. If, at a later time, the model became inaccurate due tochanges in external uncontrollable aspects of the plant 600, then themodel 610 would no longer be accurate. However, it is noted that thegain of the steady state model 610 would remain accurate due to the factthat an offset would be present. In order to account for this offset, adynamic model 612 is utilized which receives the inputs values andgenerates the output values CV which provide a prediction of the outputvalue CV. This is compared with an actual plant output value which isderived from a virtual on-line analyzer (VOA) 616. The VOA 616 isoperable to receive the plant outputs CV and the inputs u(t). The outputof the VOA 616 provides the actual output of the plant which is input adifference circuit 618, the difference thereof being the offset or“bias.” This bias represents an offset which is then filtered with afilter 620 for input to an offset device 622 to offset external CV setpoints, i.e., desired CV values, for input to the nonlinear steady statemodel 610.

Referring now to FIG. 48, there is illustrated a diagrammatic view ofthe application of the bias illustrated in FIG. 47. There is illustrateda curve 630 representing the mapping of the input space to the outputspace through a steady state model. As such, for each manipulatablevariable (MV), there will be provided a set of output variables (CV).Although the steady state model is illustrated as a straight line, itcould have a more complex surface. Also, it should be understood thatthis is represented as a single dimension, but this could equally applyto a multi-variable system wherein there are multiple input variablesfor a given input vector and multiple output values for a given outputvector. If the steady state model represents an accurate predictivemodel of the system, then input vector MV will correspond to a predictedoutput value. Alternatively, in a control environment, it is desirableto predict the MVs from a desired output value CV_(SET). However, thepredicted output value or the predicted input value, depending uponwhether the input is predicted or the output is predicted, will be afunction of the accuracy of the model. This can change due to variousexternal unmeasurable disturbances such as the external temperature, thebuildup of slag in a boiler, etc. This will effectively change the waythe plant operates and, therefore, the model will no longer be valid.However, the “gain” or sensitivity of the model should not change due tothese external disturbances. As such, when an MV is varied, the outputwould be expected to vary from an initial starting point to a finalresting point in a predictable manner. It is only the value of thepredicted value at the starting point that is in question. In order tocompensate for this, some type of bias must be determined and an offsetprovided.

In the diagrammatic view of FIG. 48, there is provided a curve 632representing the actual output of the plant as determined by the VOA616. A bias is measured at the MV point 634 such that an offset can beprovided. When this offset is provided, there will be an optimized MV,MV_(OPT). This will occur at a point 636 on the curve 632. Therefore, inorder to provide a desired value CV_(SET), the MV that must beassociated with that for the operation of the current plant must beMV_(OPT).

Referring now to FIG. 49, there is illustrated a plot of both CV and MVillustrating the dynamic operation. There is provided a first plot orcurve 640 illustrating the operation of the CV output in response to adynamic change which is the result of a step change in the MV input,represented by curve 642. The dynamic model 612 will effectively predictwhat will happen to the plant when it is not in steady state. Thisoccurs in a region 646. In the upper region, a region 648, thisconstitutes a steady state region.

Referring now to FIG. 50, there is illustrated a plot illustrating thepredicted model as being a steady state model. Again, there is providedthe actual output curve 640 which represents the output of the VOA andalso a curve 650 which represents the prediction provided by the steadystate nonlinear model. It can be seen that, when the curve 642 makes astep change, the output, as represented by the curve 640, will changegradually up to a steady state value. However, the steady state modelwill make an immediate calculation of what the steady state value shouldbe, as represented by a transition 652. This will rise immediately tothe steady state level such that, during the region 646, the predictionwill be inaccurate. This is representative in the plot of bias for boththe dynamic and the steady state configurations. In the bias for thesteady state model, represented by a solid curve 654, the steady statebias will become negative for a short period of time and then, duringthe region 646, go back to a bias equal to that of the dynamic model.The dynamic model bias is illustrated by a dotted line 656.

Referring now to FIG. 51 there are illustrated curves representing theoperation of the dynamic model wherein the dynamic model does notaccurately predict the dynamics of the system. Again, the output of thesystem, represented by the VOA, is represented by the curve 640. Thedynamic model provides a predictive output incorporated in the dynamicsof the system, which is represented by a curve 658. It can be seen that,during the dynamic portion of the curve represented by region 646, thatthe dynamic model reaches a steady state value too quickly, i.e., itdoes not accurately model the dynamics of the plant during thetransition in region 646. This is represented by a negative value in thebias, represented by curve 660 with a solid line. The filter 620 isutilized to filter out the fast transitions represented by the dynamicmodel in the output bias value (not the output of the dynamic modelitself). Also, it can be seen that the bias, represented by a dottedline 664, will be less negative.

Referring now to FIG. 52, there is illustrated a block diagram of aprior art system utilizing a steady state model for optimization. Inthis system, there is provided a steady state model 670 similar to thesteady state model 610. This is utilized to receive set points andpredict optimized MVs, represented as the vector u_(OPT). However, inorder to provide some type of bias for the operation thereof, the actualset points are input to an offset circuit 672 to be offset by a biasinput. This bias input is generated by comparing the output of a steadystate model 672, basically the same steady state model 670, with theoutput of a VOA 674, similar to VOA 616 in FIG. 47. This will provide abias value, as it does in FIG. 47. However, it is noted that this is thebias between the steady state model, a steady state predicted value, andpossibly the output of the plant which may be dynamic in nature.Therefore, it is not valid during dynamic changes of the system. It isonly valid at a steady state condition. In order to utilize the biasoutput by a difference circuit 676 which compares the output of a steadystate model 672 with that of the VOA 674, a steady state detector 678 isprovided. This steady state detector 678 will look at the inputs andoutputs of the network and determine when the outputs have “settled” toan acceptable level representative of a steady state condition. This canthen be utilized to control a latch 680 which latches the output of thedifference circuit 676, the bias value, which latched value is theninput to the offset circuit 672. It can therefore be seen that thisconfiguration can only be utilized in an off-line mode, i.e., when thereare no dynamics being exhibited by the system.

Steady State Optimization

In general, steady state models are utilized for the steady stateoptimization. Steady state models represent the steady state mappingbetween inputs of the process (manipulated variables (MV) anddisturbance variables (DV)) and outputs (controlled variables (CV)).Since the models represent a steady state mapping, each input and outputprocess is represented by a single input or output to the model (timedelays in the model are ignored for optimization purposes). In general,the gains of a steady state model must be accurate while the predictionsare not required to be accurate. Precision in the gain of the model isneeded due to the fact that the steady state model is utilized in anoptimization configuration. The steady state model need not yield anaccurate prediction due to the fact that a precise VOA can be used toproperly bias the model. Therefore, the design of the steady state modelis designed from the perspective of developing an accurate gain modelduring training thereof to capture the sensitivities of the plant. Themodel described hereinbelow for steady state optimization is a nonlinearmodel which is more desirable when operating in multiple operatingregions. Moreover, when operating in a single operating region, a linearmodel could be utilized. A single operating region process is defined asa process whose controlled variables operate at constant set-points,whose measured disturbance variables remain in the same region, andwhose manipulated variables are only changed to reject unmeasureddisturbances. Since the MVs are only moved to reject externaldisturbances, the process is defined to be external disturbancedominated. An example of a single operating region process is adistillation column. By comparison, a multiple operating region processis a process whose manipulated variables are not only moved to rejectunmeasured disturbances, but are also changed to provide desired processperformance. For example, the manipulated variables may be changed toachieve different CV set points or they may be manipulated in responseto large changes to measured disturbances. An example of this could be adistillation column with known and significant changes in feed rate orcomposition (measured disturbance variable) which operates at a constantset point. Since the MVs or CVs of a multiple operating region processare often set to non-constant references to provide desired processperformance, a multiple region process is reference dominated ratherthan disturbance dominated. The disclosed embodiment herein is directedtoward multiple operating region processes and, therefore, a non-lineardynamic model will be utilized. However, it should be understood thatboth the steady state model and the dynamic model could be linear innature to account for single operating region processes.

As described hereinabove, steady state models are generated or trainedwith the use of historical data or a first principals model, even ageneric model of the system. The MVs should be utilized as inputs to themodels but the states should not be used as inputs to the steady statemodels. Using states as inputs in the steady state models, i.e., thestates being the outputs of the plant, produces models with accuratepredictions but incorrect gains. For optimization purposes, as describedhereinabove, the gain is the primary objective for the steady stateoptimization model.

Referring now to FIG. 53, there is illustrated a block diagram of amodel which is utilized to generate residual or computed disturbancevariables (CDVs). A dynamic non-linear state model 690 which provides amodel of the states of the plant, the measurable outputs of the plant,and the MVs and DVs. Therefore, this model is trained on the dynamics ofthe measurable outputs, the states, and the MVs and DVs as inputs. Thepredicted output, if accurate, should be identical to the actual outputof the system. However, if there are some unmeasurable externaldisturbances which affect the plant, then this prediction will beinaccurate due to the fact that the plant has changed over that whichwas originally modeled; Therefore, the actual state values, the measuredoutputs of the plant, are subtracted from the predicted states toprovide a residual value in the form of the CDVs. Thereafter, thecomputed disturbances, the CDVs, are provided as an input to anon-linear steady state model 692, illustrated in FIG. 54, in additionto the MVs and DVs. This will provide a prediction of the CVs on theoutput thereof. Non-linear steady state model 692, describedhereinabove, is created with historical data wherein the states are notused as inputs. However, the CDVs provide a correction for the externaldisturbances. This is generally referred to as a residual activationnetwork which was disclosed in detail in U.S. Pat. No. 5,353,207 issuedOct. 4, 1994 to J. Keeler, E. Hartman and B. Ferguson, which patent isincorporated herein by reference.

On-Line Dynamic Optimization

Referring now to FIG. 55, there is illustrated a detailed block diagramof the model described in FIG. 47. The non-linear steady state model610, described hereinabove, is trained utilizing manipulated variables(MV) as inputs with the outputs being the CVs. In addition it alsoutilizes the DVs. The model therefore is a function of both the MVs, theDVs and also the CDVs, as follows:CV=ƒ(MV, DV, CDV)  (81)The non-linear steady state model 610 is utilized in an optimizationmode wherein a cost function is defined to which the system is optimizedsuch that the MVs can move only within the constraints of the costfunction. The cost function is defined as follows:J=ƒ(MV, DV, CV _(SET))  (82)noting that many other factors can be considered in the cost function,such as gain constrains, economic factors, etc. The optimizedmanipulatable variables (MV_(OPT)) are determined by iteratively movingthe Mvs based upon the derivative dJ\dMV. This is a conventionaloptimization technique and is described in Mash, S. G. and Sofer, A.,“Linear and Nonlinear Programming,” McGraw Hill, 1996, which isincorporated herein by reference. This conventional steady stateoptimizer is represented by a block 700 which includes the non-linearsteady state model 610 which receives both the CDVs, the DVs and a CVset point. However, the set point is offset by the offset block 672.This offset is determined utilizing a non-linear dynamic predictionnetwork comprised of the dynamic non-linear state model 690 forgenerating the CDVs, from FIG. 53, which CDVs are then input to anon-linear dynamic model 702. Therefore, the combination of the dynamicnon-linear state model 690 for generating the CDVs and the non-lineardynamic model 702 provide a dynamic prediction on output 704. This isinput to the difference circuit 618 which provides the bias for input tothe filter 620. Therefore, the output of the VOA 616 which receives bothstates as an input the MVs and DVs as inputs provides an output thatrepresents the current output of the plant. This is compared to thepredicted output and the difference thereof constitutes a bias. VOA 616can be a real time analyzer that provides an accurate representation ofthe current output of the plant. The non-linear dynamic model 702 isrelated to the non-linear steady state model 610 as describedhereinabove, in that the gains are related.

The use of the non-linear dynamic model 702 and the dynamic non-linearstate model 690 provides a dynamic representation of the plant which canbe compared to the output of the VOA 616. Therefore the bias willrepresent the dynamics of the system and, therefore, can be utilized online.

Referring now to FIG. 56, there is illustrated a diagrammatic view of afurnace/boiler system which has associated therewith multiple levels ofcoal firing. The central portion of the furnace/boiler comprises afurnace 720 which is associated with the boiler portion. The furnaceportion 720 has associated therewith a plurality of delivery ports 722spaced about the periphery of the boiler at different elevations. Eachof the delivery ports 720 has associated therewith a pulverizer 724 anda coal feeder 726. The coal feeder 726 is operable to feed coal into thepulverizer 724 at a predetermined rate. The pulverizer 724 mixes thepulverized coal with air and then injects it into the furnace portion720. The furnace/boiler will circulate the heated air through multipleboiler portions represented by a section 730 which provides variousmeasured outputs (CV) associated with the boiler operation as will bedescribed hereinbelow. In addition, the exhaust from the furnace whichis re-circulated, will have nitrous oxides (NO_(x)) associatedtherewith. An NO_(x) sensor 732 will be provided for that purpose.

Referring now to FIG. 57, there is illustrated a cross-sectional view ofthe furnace portion 720 illustrating four of the delivery ports 722spaced about the periphery of the furnace portion 720. The pulverizedcoal is directed into the furnace portion 720 to the interior thereoftangential to what is referred to as a “fireball” interior to thefurnace. This is what is referred to as a “tangentially fired” boiler.Utilizing this technique, a conventional technique, a fireball can becreated proximate to the delivery or inlet ports 722. Further, the feedrates for each of the elevations and the associated inlet ports 722 canbe controlled in order to define the center of the fireball. By varyingthe feed rates to the various elevations, this fireball can be placed atcertain levels. The placement of this fireball can have an effect on theefficiency and the NO_(x) level. For this particular application, twoimportant features to control or optimize are the efficiency and theNO_(x) levels.

In these boilers with multiple elevations of coil firing, thecombination of the elevations in service is an important parameter whenutilizing a prediction and optimization system. This is specifically sowith respect to optimizing the NO_(x) emissions and the performanceparameters. Additionally, some of these boilers in the field have excessinstalled pulverizer capacity for delivering fuel for each elevationand, therefore, opportunities exist to alter the way fuel is introducedusing any given combination, as additional fuel can be added. Ingeneral, for any given output level a relatively stable coal feed rateis required such that the increase or decrease of fuel flow to oneelevation results in a corresponding, opposite direction change in coalflow to another elevation. A typical utility boiler will have betweenfour to eight elevations of fuel firing and may have dual furnaces. Thispresents a problem in that representation of a plant in a neural networkor some type of first principals model will require the model torepresent the distribution of fuel throughout the boiler in an empiricalmodel with between four and sixteen highly correlated, coal flow inputvariables. Neural networks, for example, being nonlinear in nature, willbe more difficult to train with so many variables.

Much of the effect on the NO_(x) emissions and performance parameters,due to these changes in fuel distribution, relate to relative height inthe boiler that the fuel is introduced. Concentrating the fuel in thebottom of the furnace by favoring the lower elevations of coal firingwill yield different output results than that concentrating the fuel atthe top of the furnace. The concept of Fuel Elevation has been developedin order to represent the relative elevation of the fuel in the furnaceas a function of the feed rate and the elevation level. This provides asingle parameter or manipulatable variable for input to the networkwhich is actually a function of multiple elevations and feed rate. TheFuel Elevation is defined as a parameter that increases between “0” and“1” as fuel is introduced higher in the furnace. Fuel Elevation iscalculated according to the following equation:

$\begin{matrix}{{Fe} = \frac{{\left( K_{1} \right)\left( R_{1} \right)} + {\left( K_{2} \right)\left( R_{2} \right)} + {\left( K_{3} \right)\left( K_{4} \right)\mspace{11mu}\ldots} + {\left( K_{n} \right)\left( R_{n} \right)}}{R_{1} + R_{2} + {R_{3}\mspace{11mu}\ldots} + R_{n}}} & (83)\end{matrix}$

=

Calculated Fuel Elevation

-   -   K₁ . . . K_(n)=Elevation Constant for elevation 1 to n        (described hereinbelow)    -   R₁ . . . R_(n)=Coal Feed Rate for elevation 1 to n.        Constants for each elevation are calculated to represent the        relative elevation that the coal from each elevation is        introduced. For example, for a unit with four elevations of fuel        firing, there are four compartments each representing 25% of the        combustion area. If the fuel is introduced to the furnace at the        center of the each of the combustion areas, then the Fuel        Elevations constants for the lowest to the highest elevations        are 0.125, 0.375, 0.625 and 0.875.

Referring now to FIG. 58, there is illustrated a block diagram of theboiler/furnace 720 connected in a feedback or control configurationutilizing an optimizer 740 in the feedback loop. A controller 742 isprovided, which controller 742 is operable to generate the variousmanipulatable variables of the input 744. The manipulatable variables,or MVs, are utilized to control the operation of the boiler. The boilerwill provide multiple measurable outputs on an output 746 referred to asthe CVs of the system or the variables to be controlled. In accordancewith the disclosed embodiment, some of the outputs will be input to theoptimizer 740 in addition to some of the inputs.

There are some inputs that will be directly input to the optimizer 740,those represented by a vector input 748. However, there are a pluralityof other inputs, represented by input vector 750, which are combined viaa multiple MV-single MV algorithm 752 for input to the optimizer 740.This algorithm 752 is operable to reduce the number of inputs andutilize a representation of the relationship of the input values to somedesired result associated with those inputs as a group. In the disclosedembodiment, this is the Fuel Elevation. This, therefore, results in asingle input on a line 754 or a reduced set of inputs.

The optimizer 740 is operable to receive a target CV on a vector input756 and also various constraints on input 758. These constraints areutilized by the optimizer, as described hereinabove. This will provide aset of optimized MVs. Some of these MVs can be directly input to thecontroller, those that are of course correlated to the input vector 748.The input vector or MV corresponding to the vector input 754 will bepassed through a single MV-multiple MV algorithm 760. This algorithm 760is basically the inverse of the algorithm 752. In general, this willrepresent the above-noted Fuel Elevation calculation. However, it shouldbe recognized that the algorithm 752 could be represented by a neuralnetwork or some type of model, as could the algorithm 760, in additionalto some type of empirical model. Therefore, the multiple inputs need tobe reduced to a lessor number of inputs or single input via some type offirst principals algorithm or some type of predetermined relationship.In order to provide these inputs to the boiler, they must be processedthrough the inverse relationship. It is important to note, as describedhereinabove, that the optimizer 740 will operate on-line, since it takesinto account the dynamics of the system.

Referring now to FIG. 59, there is illustrated a block diagram of thetraining system for training the optimizer 740 at the neural networks. Ageneral model 770 is provided which is any type of trainable nonlinearmodel. These models are typically trained via some type ofbackpropagation technique such that a training database is required,this represented by training database 772. A training system 774operates the model 770 such that it is trained on the outputs and theinputs. Therefore, inputs are applied thereto with target outputsrepresenting the plant. The weights are adjusted in the model through aniterative procedure until the error between the outputs and the inputsis minimized. This, again, is a conventional technique.

In the disclosed embodiment, since there is defined a relationshipbetween multiple inputs to a single or reduced set of inputs, it isnecessary to train the model 770 with this relationship in place.Therefore, the algorithm 752 is required to reduce the plurality ofinputs on vector 750 to a reduce set of inputs on the vector 754, which,in the disclosed embodiment, is a single value. This will constitute asingle input or a reduced set of inputs that replace the multiple inputson vector 750, which input represents some functional relationshipbetween the inputs and some desired or observed behavior of the boiler.Thereafter, the remaining inputs are applied to the model 770 in theform of the vector 748. Therefore, once the model 770 is trained, it istrained on the representation generated by the algorithm in the multipleMV-single MV algorithm 752.

Referring now to FIGS. 60 and 61, there is illustrated a more detailedblock diagram of the embodiment of FIG. 57. The portion of theembodiment illustrated in FIG. 59 is directed toward that necessary togenerate the CV bias for biasing the set points. There are a pluralityof measured inputs (MV) that are provided for the conventional boiler.These are the feeder speeds for each of the pulverizers, the CloseCoupled Over-fired Air (CCOFA) value comprising various vents or dampersthat have a preset open value, a Separated Over-fired Air (SOFA) valuewhich also is represented in terms of a percent open of select dampers,a tilt value, which defined the tilt of the inlet ports for injectingthe fuel, the Wind Box to Furnace Pressure (WB/Fur) which all areutilized to generate input variables for a network. With respect tooutputs, the outputs will be in the form of the NO_(x) values determinedby a sensor: the dry gas loss, the main steam temperature, and the losson ignition (LOI) for both reheat and superheat.

Of the inputs, the feeder speeds are input to a Fuel Elevation algorithmblock 790 which provides a single output on an output 792 which isreferred to as the Fuel Elevation concept, a single value. In addition,the multiple feeder speeds are input to an auxiliary air elevationalgorithm block 794, which also provides a single value representingauxiliary air elevation, this not being described in detail herein, itbeing noted that this again is a single value representing arelationship between multiple inputs and a desired parameter of theboiler. The CCOFA values for each of the dampers provide arepresentation of a total CCOFA value and a fraction CCOFA value, andrepresented by an algorithm block 796. This is also the case with theSOFA representation in a block 798 and also with the WB/Furrepresentation wherein a pseudo curve is utilized and a delta value isdetermined from that pseudo curve based upon the multiple inputs. Thisis represented by a block 800. The output of all of the blocks 790, 794,796, 798, and 800 provide the MVs in addition to the Stack O₂ value on aline 802. These are all input to a state prediction model 804 similar tothe model 690 in FIG. 55. This model also receives the disturbancevariables (DV) for the system, these not being manipulatable inputs tothe boiler. The state prediction then provides the predicted states to adifference block 806 for determining the difference between predictedstates and the measured state variable output by the boiler 720 on aline 808 which provides the measured states. This is the current outputof the boiler 720. The difference block 806 provides the ComputedDisturbance Variables (CDV) which are filtered in a filter 810. Thisessentially, for the boiler, will be due to the slag and the cleanlinessof the boiler. Thereafter, the CDVs, the MVs and the DVs are input to amodel 812, which basically is the nonlinear dynamic model 702. Theoutput of model 812 provides the estimated CVs which are then comparedwith the measured CVs in a difference block 814 to provide through afilter 816 the CV bias value.

Referring further to FIG. 60, the CV bias value output by the filter 816is input to a steady state model optimization block 818, in addition tothe MVs, DVs, and the CDVs. In addition, there are provided varioustargets in the form of set points, various optimization weightings andvarious constraints such as gain constraints, etc. This is substantiallyidentical to the steady state optimizer 700 illustrated in FIG. 55. Itshould be understood that this is a conventional optimization techniquewhich defines a cost function which operates on the targets,constraints, and the optimization weightings in the cost functionrelationship and then essentially utilizes the derivative of the costfunction to determine the move of the MV. This is conventional and isdescribed in Mash, S. G. and Sofer, A., “Linear and NonlinearProgramming,” McGraw Hill, 1996, which was incorporated herein byreference.

The optimization model 818 will provide Mv set points. These MV setpoints could be, for such MVs as the Stack O₂, input directly to theboiler 720 for control thereof as a new input value. However, when theMVs that represent the single values such as Fuel Elevation whichrelates back to multiple inputs must be processed through the inverse ofthat relationship to generate the multiple inputs. For example, FuelElevation value is provided as Mv on a line 820 for input to a FuelElevation neural network 822 which models the relationship of FuelElevation to feeder speed. However, a neural network is not necessarilyrequired in that the basic relationship described hereinabove withrespect to Fuel Elevation will suffice and the algorithm required isonly the inverse of that relationship. This will provide on the outputfeeder speeds on lines 824, which are multiple inputs. In addition, theauxiliary elevation is processed through a representation of a neuralnetwork which relates the multiple auxiliary air open values to the MVinput in a block 826. The CCOFA representation provides the same inverserelationship in a block 828 to provide the MV set points associatedtherewith, the CCOFA, and the fraction CCOFA and provide the CCOFApercent open values. Of the SOFA MVs, the total SOFA and the fractionSOFA are processed through an inverse SOFA representation to provide theSOFA percent open inputs to the boiler 720. Lastly, the delta value fromthe WB/Fur curve is provided as MV set point through an inverserelationship in a block 832 in order to determine the WB/Fur input valueto the boiler 720. All of these operations, the optimization and theconversion operations, are done in real time, such that they must takeinto account the dynamics of this system. Further, as describedhereinabove, by reducing the amount of inputs, the actual steady statemodels and dynamic models will provide a better representation, and thesensitivities have been noted as being augmented for theserepresentations. With this technique, the center of mass of the ball inthe furnace 720 can be positioned with the use of one representativeinput modeled in the neural network or similar type model to allowefficiency and NOx to be optimized. It is noted that each of the inputsthat represents multiple inputs to any of the algorithm blocks notedhereinabove and which are represented by a single variable each have apredetermined relationship to each other, i.e., the feed rate at eachelevation has some relationship to the other elevations only withrespect to a parameter defined as the center of mass of the fireball.Otherwise, each of these feeder rates is independent of each other. Bydefining a single parameter that is of interest and the relationshipbetween the inputs to define this relationship, then that parameteritself can become a more important aspect of the model.

In summary, there has been provided an on-line optimizer which providesan estimation of a plant which can then be compared to the actual outputof the plant. This difference creates a bias which bias will then beutilized to create an offset to target set points for the optimizationprocess. Therefore, the steady state optimizer, which can include anonlinear steady model for a multi-variable system can be optimized forgain as opposed to an accurate prediction. A nonlinear dynamic modelprovides for an accurate prediction and also provides an estimation ofthe dynamics of the system. Therefore, the bias will have reflectedtherein dynamics of the system.

Although the preferred embodiment has been described in detail, itshould be understood that various changes, substitutions and alterationscan be made therein without departing from the spirit and scope of theinvention as defined by the appended claims.

Appendix: The Second Derivatives

Gradient descent (equation 66) requires the second derivatives

$\begin{matrix}{\frac{\partial}{\partial w}\left( \frac{\partial y_{k}}{\partial x_{i}} \right)} & (84)\end{matrix}$where w denotes an arbitrary weight or bias in the network. ThisAppendix gives the expressions for these second derivatives.

Direct input-output connections are often necessary in realapplications. Generalizing the recursive equations of (lee and Oh 1007)to include layer-skipping connections would be relativelystraight-forward. Instead, the second derivatives are provided in aneasy-to-implement nonrecursive form for the usual three-layer network,and include direct input-output connections.

Full connectivity between layers is not assumed. A second derivative∂/∂w (∂y_(k)/∂x_(i)) is zero if no path exists from w to y_(k). Thequestions below are valid provided weights corresponding to non-existingconnections are fixed at zero.

Let a_(k) and a_(j) denote the total input to the model to output y_(k)and to hidden layer h_(j) respectively:

$\begin{matrix}{a_{k} = {{\sum\limits_{j}{w_{kj}h_{j}}} + {\sum\limits_{i}{w_{ki}x_{i}}} + b_{k}}} & (85) \\{a_{j} = {{\sum\limits_{i}{w_{ji}x_{i}}} + b_{j}}} & (86)\end{matrix}$The activation values are theny _(k) ^(σ)=σ(a _(k))y _(k) ^(L) =a _(k)  (87)h _(j) ^(σ)=σ(a _(j))h _(j) ^(L) =a _(j)  (88)where the superscripts a and L indicate sigmoid and linear units,respectively, and a(.) Is the usual sigmoid transfer functiona(a)=(1+e^(−a))⁻¹. The derivatives for output and hidden units are then

$\begin{matrix}{{y_{k}^{\prime} \equiv \frac{\mathbb{d}y_{k}}{\mathbb{d}a_{k}}} = \begin{Bmatrix}{y_{k}\left( {1 - y_{k}} \right)} & {{for}\mspace{14mu} y_{k}^{\sigma}} \\1 & {{for}\mspace{14mu} y_{k}^{L}}\end{Bmatrix}} & (89) \\{{h_{j}^{\prime} \equiv \frac{\mathbb{d}h_{j}}{\mathbb{d}a_{j}}} = \begin{Bmatrix}{h_{j}\left( {1 - h_{j}} \right)} & {{for}\mspace{14mu} h_{j}^{\sigma}} \\1 & {{for}\mspace{14mu} h_{j}^{L}}\end{Bmatrix}} & (90)\end{matrix}$For conciseness in the equations below, we define the quantities

$\begin{matrix}{u_{k} = \begin{Bmatrix}{1 - {2y_{k}}} & {{for}\mspace{14mu} y_{k}^{\sigma}} \\0 & {{for}\mspace{14mu} y_{k}^{L}}\end{Bmatrix}} & (100) \\{u_{j} = \begin{Bmatrix}{1 - {2h_{j}}} & {{for}\mspace{14mu} h_{j}^{\sigma}} \\0 & {{for}\mspace{14mu} h_{j}^{L}}\end{Bmatrix}} & (101)\end{matrix}$The gains ∂y_(k)/∂x_(i) appearing in the equations below are computedaccording to the above described section entitled “Training Procedure.”The second derivative equations are written separately for each type ofweight and bias:

Hidden-output weight:

$\begin{matrix}{{\frac{\partial}{\partial w_{kj}}\left( \frac{\partial y_{m}}{\partial x_{n}} \right)} = {\delta_{km}\left( {{u_{m}h_{j}\frac{\partial y_{m}}{\partial x_{n}}} + {y_{m}^{\prime}h_{j}^{\prime}w_{jn}}} \right)}} & (102)\end{matrix}$

Where ∂_(km) is a Kronecker delta.

$\begin{matrix}{{\bullet\mspace{14mu}{Input}\text{-}{output}\mspace{14mu}{weight}\text{:}}\mspace{20mu}{{\frac{\partial}{\partial w_{ki}}\left( \frac{\partial y_{m}}{\partial x_{n}} \right)} = {\delta_{km}\left( {{u_{m}x_{i}\frac{\partial y_{m}}{\partial x_{n}}} + {y_{m}^{\prime}\delta_{in}}} \right)}}} & (103) \\{{\bullet\mspace{14mu}{Input}\text{-}{hidden}\mspace{14mu}{weight}\text{:}}\mspace{20mu}{{\frac{\partial}{\partial w_{ji}}\left( \frac{\partial y_{m}}{\partial x_{n}} \right)} = {h_{j}^{\prime}{w_{mj}\left\lbrack {{u_{m}x_{i}\frac{\partial y_{m}}{\partial x_{n}}} + {y_{m}^{\prime}\left( {{u_{j}x_{i}w_{jn}} + \delta_{in}} \right)}} \right\rbrack}}}} & (104) \\{{\bullet\mspace{14mu}{Output}\mspace{14mu}{bias}\text{:}}\mspace{20mu}{{\frac{\partial}{\partial b_{k}}\left( \frac{\partial y_{m}}{\partial x_{n}} \right)} = {\delta_{km}\left( {u_{m}\frac{\partial y_{m}}{\partial x_{n}}} \right)}}} & (105) \\{{\bullet\mspace{14mu}{Hidden}\mspace{14mu}{bias}\text{:}}\mspace{20mu}{{\frac{\partial}{\partial b_{j}}\left( \frac{\partial y_{m}}{\partial x_{n}} \right)} = {h_{j}^{\prime}{w_{mj}\left( {{u_{m}\frac{\partial y_{m}}{\partial x_{n}}} + {y_{m}^{\prime}u_{j}w_{jn}}} \right)}}}} & (106)\end{matrix}$

1. A method, comprising: providing a model, wherein the model comprisesa representation of a plant or process, wherein the model comprises oneor more inputs and one or more outputs, wherein the model comprises oneor more gains, and wherein each gain comprises a respective partialderivative of an output with respect to a respective input; specifyingone or more user-configurable gain constraints, wherein each of the oneor more user-configurable gain constraints corresponds to a respectivegain; and training the model with a non-linear programming optimizersubject to the one or more user-configurable gain constraints.
 2. Themethod of claim 1, wherein said training the model comprises: a)receiving training input data representative of inputs to the plant orprocess; b) executing the model to generate predicted training outputdata for the plant or process corresponding to the training input data;c) comparing the predicted training output data to training output datarepresentative of output of the plant or process to determine an error;d) updating the model in accordance with the determined error; and e)performing a)-d) in an iterative manner to minimize the error.
 3. Themethod of claim 2, wherein said training the model further comprisesoptimizing a specified objective function.
 4. The method of claim 1,wherein the one or more user-configurable gain constraints comprise oneor more of: at least one hard constraint, comprising a strict limitationon a corresponding at least one gain used in said training the model; orat least one soft constraint, comprising a weighted penalty with respectto a corresponding at least one gain used in said training the model. 5.The method of claim 1, wherein at least one of the user-configurablegain constraints comprises one or more of: a lower bound on thecorresponding gain; or an upper bound on the corresponding gain.
 6. Themethod of claim 1, wherein the model comprises a neural network.
 7. Themethod of claim 1, the method further comprising: providing plant orprocess input data to the model; executing the model to generatepredicted plant or process output based on the plant or process inputdata; and wherein the predicted plant or process output is useable tomanage the plant or process.
 8. The method of claim 1, the methodfurther comprising: specifying a desired behavior of the plant orprocess; optimizing the model with one or more other optimizers todetermine one or more input values that result in the desired behaviorof the plant or process; and wherein the one or more input values areusable to operate the plant or process to produce the desired behavior.9. The method of claim 8, further comprising: providing the one or moreinput values as input to the plant or process; operating the plant orprocess in accordance with the one or more input values to produce thedesired behavior.
 10. The method of claim 8, wherein the one or moreother optimizers comprise the non-linear programming optimizer.
 11. Themethod of claim 8, the method further comprising: the one or more otheroptimizers optimizing a specified objective function with respect to themodel.
 12. The method of claim 11, wherein said optimizing the specifiedobjective function comprises: optimizing the specified objectivefunction subject to one or more constraints.
 13. The method of claim 12,wherein the one or more constraints comprise one or more of: at leastone hard constraint, wherein the at least one hard constraint isstrictly satisfied during said optimization; or at least one softconstraint, comprising a weighted penalty used in said optimizing thespecified objective function.
 14. The method of claim 12, wherein theone or more constraints comprise one or more soft constraints, andwherein the one or more other optimizers are operable to optimize theplant or process via gradient descent.
 15. The method of claim 1,wherein the model comprises a steady state model.
 16. The method ofclaim 1, wherein the model comprises a dynamic model.
 17. The method ofclaim 1, wherein said training comprises: training the model using oneor more of: one or more other constraints; or one or more otheroptimizers.
 18. A computer-accessible memory medium, wherein the memorymedium comprises program instructions executable to: implement a model,wherein the model comprises one or more inputs and one or more outputs,wherein the model comprises a representation of a plant or process,wherein the model comprises one or more gains, and wherein each gaincomprises a respective partial derivative of an output with respect to arespective input; specify one or more user-configurable gainconstraints, wherein each of the one or more user-configurable gainconstraints corresponds to a respective gain; and training the modelwith a non-linear programming optimizer subject to the one or moreuser-configurable gain constraints.
 19. A system, comprising: means forproviding a model, wherein the model comprises one or more inputs andone or more outputs, wherein the model comprises a representation of aplant or process, wherein the model comprises one or more gains, andwherein each gain comprises a respective partial derivative of an outputwith respect to a respective input; means for specifying one or moreuser-configurable gain constraints, wherein each of the one or moreuser-configurable gain constraints corresponds to a respective gain; andtraining the model with a non-linear programming optimizer subject tothe one or more user-configurable gain constraints.
 20. A system,comprising: a model, representing a plant or process, the modelcomprising: one or more inputs; one or more outputs; and one or moregains, wherein each gain comprises a respective partial derivative of anoutput with respect to a respective input; and a non-linear programmingoptimizer, coupled to the model; wherein the model is constrained by oneor more user-configurable gain constraints, wherein each of the one ormore user-configurable gain constraints corresponds to a respectivegain; and wherein the non-linear programming optimizer is operable totrain the model subject to the one or more user-configurable gainconstraints.